:: LIMFUNC1 semantic presentation begin Lm1: for g, r, r1 being real number st 0 < g & r <= r1 holds ( r - g < r1 & r < r1 + g ) proof let g, r, r1 be real number ; ::_thesis: ( 0 < g & r <= r1 implies ( r - g < r1 & r < r1 + g ) ) assume A1: ( 0 < g & r <= r1 ) ; ::_thesis: ( r - g < r1 & r < r1 + g ) then r - g < r1 - 0 by XREAL_1:15; hence r - g < r1 ; ::_thesis: r < r1 + g r + 0 < r1 + g by A1, XREAL_1:8; hence r < r1 + g ; ::_thesis: verum end; Lm2: for seq being Real_Sequence for f1, f2 being PartFunc of REAL,REAL st rng seq c= dom (f1 + f2) holds ( dom (f1 + f2) = (dom f1) /\ (dom f2) & rng seq c= dom f1 & rng seq c= dom f2 ) proof let seq be Real_Sequence; ::_thesis: for f1, f2 being PartFunc of REAL,REAL st rng seq c= dom (f1 + f2) holds ( dom (f1 + f2) = (dom f1) /\ (dom f2) & rng seq c= dom f1 & rng seq c= dom f2 ) let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( rng seq c= dom (f1 + f2) implies ( dom (f1 + f2) = (dom f1) /\ (dom f2) & rng seq c= dom f1 & rng seq c= dom f2 ) ) assume A1: rng seq c= dom (f1 + f2) ; ::_thesis: ( dom (f1 + f2) = (dom f1) /\ (dom f2) & rng seq c= dom f1 & rng seq c= dom f2 ) thus dom (f1 + f2) = (dom f1) /\ (dom f2) by VALUED_1:def_1; ::_thesis: ( rng seq c= dom f1 & rng seq c= dom f2 ) then ( dom (f1 + f2) c= dom f1 & dom (f1 + f2) c= dom f2 ) by XBOOLE_1:17; hence ( rng seq c= dom f1 & rng seq c= dom f2 ) by A1, XBOOLE_1:1; ::_thesis: verum end; Lm3: for seq being Real_Sequence for f1, f2 being PartFunc of REAL,REAL st rng seq c= dom (f1 (#) f2) holds ( dom (f1 (#) f2) = (dom f1) /\ (dom f2) & rng seq c= dom f1 & rng seq c= dom f2 ) proof let seq be Real_Sequence; ::_thesis: for f1, f2 being PartFunc of REAL,REAL st rng seq c= dom (f1 (#) f2) holds ( dom (f1 (#) f2) = (dom f1) /\ (dom f2) & rng seq c= dom f1 & rng seq c= dom f2 ) let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( rng seq c= dom (f1 (#) f2) implies ( dom (f1 (#) f2) = (dom f1) /\ (dom f2) & rng seq c= dom f1 & rng seq c= dom f2 ) ) assume A1: rng seq c= dom (f1 (#) f2) ; ::_thesis: ( dom (f1 (#) f2) = (dom f1) /\ (dom f2) & rng seq c= dom f1 & rng seq c= dom f2 ) thus dom (f1 (#) f2) = (dom f1) /\ (dom f2) by VALUED_1:def_4; ::_thesis: ( rng seq c= dom f1 & rng seq c= dom f2 ) then ( dom (f1 (#) f2) c= dom f1 & dom (f1 (#) f2) c= dom f2 ) by XBOOLE_1:17; hence ( rng seq c= dom f1 & rng seq c= dom f2 ) by A1, XBOOLE_1:1; ::_thesis: verum end; notation let r be real number ; synonym left_open_halfline r for halfline r; end; definition let r be real number ; func left_closed_halfline r -> Subset of REAL equals :: LIMFUNC1:def 1 ].-infty,r.]; coherence ].-infty,r.] is Subset of REAL proof for x being set st x in ].-infty,r.] holds x in REAL by XREAL_0:def_1; hence ].-infty,r.] is Subset of REAL by TARSKI:def_3; ::_thesis: verum end; func right_closed_halfline r -> Subset of REAL equals :: LIMFUNC1:def 2 [.r,+infty.[; coherence [.r,+infty.[ is Subset of REAL proof for x being set st x in [.r,+infty.[ holds x in REAL by XREAL_0:def_1; hence [.r,+infty.[ is Subset of REAL by TARSKI:def_3; ::_thesis: verum end; func right_open_halfline r -> Subset of REAL equals :: LIMFUNC1:def 3 ].r,+infty.[; coherence ].r,+infty.[ is Subset of REAL proof for x being set st x in ].r,+infty.[ holds x in REAL by XREAL_0:def_1; hence ].r,+infty.[ is Subset of REAL by TARSKI:def_3; ::_thesis: verum end; end; :: deftheorem defines left_closed_halfline LIMFUNC1:def_1_:_ for r being real number holds left_closed_halfline r = ].-infty,r.]; :: deftheorem defines right_closed_halfline LIMFUNC1:def_2_:_ for r being real number holds right_closed_halfline r = [.r,+infty.[; :: deftheorem defines right_open_halfline LIMFUNC1:def_3_:_ for r being real number holds right_open_halfline r = ].r,+infty.[; theorem :: LIMFUNC1:1 for seq being Real_Sequence holds ( ( seq is non-decreasing implies seq is bounded_below ) & ( seq is non-increasing implies seq is bounded_above ) ) proof let seq be Real_Sequence; ::_thesis: ( ( seq is non-decreasing implies seq is bounded_below ) & ( seq is non-increasing implies seq is bounded_above ) ) thus ( seq is non-decreasing implies seq is bounded_below ) ::_thesis: ( seq is non-increasing implies seq is bounded_above ) proof assume A1: seq is non-decreasing ; ::_thesis: seq is bounded_below take (seq . 0) - 1 ; :: according to SEQ_2:def_4 ::_thesis: for b1 being Element of NAT holds not seq . b1 <= (seq . 0) - 1 let n be Element of NAT ; ::_thesis: not seq . n <= (seq . 0) - 1 ( (seq . 0) - 1 < (seq . 0) - 0 & seq . 0 <= seq . n ) by A1, SEQM_3:11, XREAL_1:15; hence not seq . n <= (seq . 0) - 1 by XXREAL_0:2; ::_thesis: verum end; assume A2: seq is non-increasing ; ::_thesis: seq is bounded_above take (seq . 0) + 1 ; :: according to SEQ_2:def_3 ::_thesis: for b1 being Element of NAT holds not (seq . 0) + 1 <= seq . b1 let n be Element of NAT ; ::_thesis: not (seq . 0) + 1 <= seq . n ( (seq . 0) + 0 < (seq . 0) + 1 & seq . n <= seq . 0 ) by A2, SEQM_3:12, XREAL_1:8; hence not (seq . 0) + 1 <= seq . n by XXREAL_0:2; ::_thesis: verum end; theorem Th2: :: LIMFUNC1:2 for seq being Real_Sequence st seq is non-zero & seq is convergent & lim seq = 0 & seq is non-decreasing holds for n being Element of NAT holds seq . n < 0 proof let seq be Real_Sequence; ::_thesis: ( seq is non-zero & seq is convergent & lim seq = 0 & seq is non-decreasing implies for n being Element of NAT holds seq . n < 0 ) assume that A1: seq is non-zero and A2: ( seq is convergent & lim seq = 0 ) and A3: seq is non-decreasing and A4: not for n being Element of NAT holds seq . n < 0 ; ::_thesis: contradiction consider n being Element of NAT such that A5: not seq . n < 0 by A4; now__::_thesis:_contradiction percases ( 0 < seq . n or seq . n = 0 ) by A5; supposeA6: 0 < seq . n ; ::_thesis: contradiction then consider n1 being Element of NAT such that A7: for m being Element of NAT st n1 <= m holds abs ((seq . m) - 0) < seq . n by A2, SEQ_2:def_7; abs ((seq . (n1 + n)) - 0) < seq . n by A7, NAT_1:12; then ( n <= n1 + n & seq . (n1 + n) < seq . n ) by A6, ABSVALUE:def_1, NAT_1:12; hence contradiction by A3, SEQM_3:6; ::_thesis: verum end; suppose seq . n = 0 ; ::_thesis: contradiction hence contradiction by A1, SEQ_1:5; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; theorem Th3: :: LIMFUNC1:3 for seq being Real_Sequence st seq is non-zero & seq is convergent & lim seq = 0 & seq is non-increasing holds for n being Element of NAT holds 0 < seq . n proof let seq be Real_Sequence; ::_thesis: ( seq is non-zero & seq is convergent & lim seq = 0 & seq is non-increasing implies for n being Element of NAT holds 0 < seq . n ) assume that A1: seq is non-zero and A2: ( seq is convergent & lim seq = 0 ) and A3: seq is non-increasing and A4: not for n being Element of NAT holds 0 < seq . n ; ::_thesis: contradiction consider n being Element of NAT such that A5: not 0 < seq . n by A4; now__::_thesis:_contradiction percases ( seq . n < 0 or seq . n = 0 ) by A5; supposeA6: seq . n < 0 ; ::_thesis: contradiction then - 0 < - (seq . n) by XREAL_1:24; then consider n1 being Element of NAT such that A7: for m being Element of NAT st n1 <= m holds abs ((seq . m) - 0) < - (seq . n) by A2, SEQ_2:def_7; A8: abs ((seq . (n1 + n)) - 0) < - (seq . n) by A7, NAT_1:12; A9: n <= n1 + n by NAT_1:12; then seq . (n1 + n) < 0 by A3, A6, SEQM_3:8; then - (seq . (n1 + n)) < - (seq . n) by A8, ABSVALUE:def_1; then seq . n < seq . (n1 + n) by XREAL_1:24; hence contradiction by A3, A9, SEQM_3:8; ::_thesis: verum end; suppose seq . n = 0 ; ::_thesis: contradiction hence contradiction by A1, SEQ_1:5; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; theorem Th4: :: LIMFUNC1:4 for seq being Real_Sequence st seq is convergent & 0 < lim seq holds ex n being Element of NAT st for m being Element of NAT st n <= m holds 0 < seq . m proof let seq be Real_Sequence; ::_thesis: ( seq is convergent & 0 < lim seq implies ex n being Element of NAT st for m being Element of NAT st n <= m holds 0 < seq . m ) assume that A1: seq is convergent and A2: 0 < lim seq and A3: for n being Element of NAT ex m being Element of NAT st ( n <= m & not 0 < seq . m ) ; ::_thesis: contradiction consider n being Element of NAT such that A4: for m being Element of NAT st n <= m holds abs ((seq . m) - (lim seq)) < lim seq by A1, A2, SEQ_2:def_7; consider m being Element of NAT such that A5: n <= m and A6: not 0 < seq . m by A3; A7: abs ((seq . m) - (lim seq)) < lim seq by A4, A5; now__::_thesis:_contradiction percases ( seq . m < 0 or seq . m = 0 ) by A6; supposeA8: seq . m < 0 ; ::_thesis: contradiction then - ((seq . m) - (lim seq)) < lim seq by A2, A7, ABSVALUE:def_1; then (lim seq) - (seq . m) < lim seq ; then lim seq < (lim seq) + (seq . m) by XREAL_1:19; then (lim seq) - (lim seq) < seq . m by XREAL_1:19; hence contradiction by A8; ::_thesis: verum end; suppose seq . m = 0 ; ::_thesis: contradiction then abs (- (lim seq)) < lim seq by A7; then abs (lim seq) < lim seq by COMPLEX1:52; hence contradiction by A2, ABSVALUE:def_1; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; theorem Th5: :: LIMFUNC1:5 for seq being Real_Sequence st seq is convergent & 0 < lim seq holds ex n being Element of NAT st for m being Element of NAT st n <= m holds (lim seq) / 2 < seq . m proof let seq be Real_Sequence; ::_thesis: ( seq is convergent & 0 < lim seq implies ex n being Element of NAT st for m being Element of NAT st n <= m holds (lim seq) / 2 < seq . m ) assume that A1: seq is convergent and A2: 0 < lim seq ; ::_thesis: ex n being Element of NAT st for m being Element of NAT st n <= m holds (lim seq) / 2 < seq . m reconsider s1 = NAT --> ((lim seq) / 2) as Real_Sequence ; A3: seq - s1 is convergent by A1; s1 . 0 = (lim seq) / 2 by FUNCOP_1:7; then lim (seq - s1) = (((lim seq) / 2) + ((lim seq) / 2)) - ((lim seq) / 2) by A1, SEQ_4:42 .= (lim seq) / 2 ; then consider n being Element of NAT such that A4: for m being Element of NAT st n <= m holds 0 < (seq - s1) . m by A2, A3, Th4, XREAL_1:215; take n ; ::_thesis: for m being Element of NAT st n <= m holds (lim seq) / 2 < seq . m let m be Element of NAT ; ::_thesis: ( n <= m implies (lim seq) / 2 < seq . m ) assume n <= m ; ::_thesis: (lim seq) / 2 < seq . m then 0 < (seq - s1) . m by A4; then 0 < (seq . m) - (s1 . m) by RFUNCT_2:1; then 0 < (seq . m) - ((lim seq) / 2) by FUNCOP_1:7; then 0 + ((lim seq) / 2) < seq . m by XREAL_1:20; hence (lim seq) / 2 < seq . m ; ::_thesis: verum end; definition let seq be Real_Sequence; attrseq is divergent_to+infty means :Def4: :: LIMFUNC1:def 4 for r being Real ex n being Element of NAT st for m being Element of NAT st n <= m holds r < seq . m; attrseq is divergent_to-infty means :Def5: :: LIMFUNC1:def 5 for r being Real ex n being Element of NAT st for m being Element of NAT st n <= m holds seq . m < r; end; :: deftheorem Def4 defines divergent_to+infty LIMFUNC1:def_4_:_ for seq being Real_Sequence holds ( seq is divergent_to+infty iff for r being Real ex n being Element of NAT st for m being Element of NAT st n <= m holds r < seq . m ); :: deftheorem Def5 defines divergent_to-infty LIMFUNC1:def_5_:_ for seq being Real_Sequence holds ( seq is divergent_to-infty iff for r being Real ex n being Element of NAT st for m being Element of NAT st n <= m holds seq . m < r ); theorem :: LIMFUNC1:6 for seq being Real_Sequence st ( seq is divergent_to+infty or seq is divergent_to-infty ) holds ex n being Element of NAT st for m being Element of NAT st n <= m holds seq ^\ m is non-zero proof let seq be Real_Sequence; ::_thesis: ( ( seq is divergent_to+infty or seq is divergent_to-infty ) implies ex n being Element of NAT st for m being Element of NAT st n <= m holds seq ^\ m is non-zero ) assume A1: ( seq is divergent_to+infty or seq is divergent_to-infty ) ; ::_thesis: ex n being Element of NAT st for m being Element of NAT st n <= m holds seq ^\ m is non-zero now__::_thesis:_ex_n_being_Element_of_NAT_st_ for_m_being_Element_of_NAT_st_n_<=_m_holds_ seq_^\_m_is_non-zero percases ( seq is divergent_to+infty or seq is divergent_to-infty ) by A1; suppose seq is divergent_to+infty ; ::_thesis: ex n being Element of NAT st for m being Element of NAT st n <= m holds seq ^\ m is non-zero then consider n being Element of NAT such that A2: for m being Element of NAT st n <= m holds 0 < seq . m by Def4; take n = n; ::_thesis: for m being Element of NAT st n <= m holds seq ^\ m is non-zero let m be Element of NAT ; ::_thesis: ( n <= m implies seq ^\ m is non-zero ) assume A3: n <= m ; ::_thesis: seq ^\ m is non-zero now__::_thesis:_for_k_being_Element_of_NAT_holds_0_<>_(seq_^\_m)_._k let k be Element of NAT ; ::_thesis: 0 <> (seq ^\ m) . k 0 < seq . (k + m) by A2, A3, NAT_1:12; hence 0 <> (seq ^\ m) . k by NAT_1:def_3; ::_thesis: verum end; hence seq ^\ m is non-zero by SEQ_1:5; ::_thesis: verum end; suppose seq is divergent_to-infty ; ::_thesis: ex n being Element of NAT st for m being Element of NAT st n <= m holds seq ^\ m is non-zero then consider n being Element of NAT such that A4: for m being Element of NAT st n <= m holds seq . m < 0 by Def5; take n = n; ::_thesis: for m being Element of NAT st n <= m holds seq ^\ m is non-zero let m be Element of NAT ; ::_thesis: ( n <= m implies seq ^\ m is non-zero ) assume A5: n <= m ; ::_thesis: seq ^\ m is non-zero now__::_thesis:_for_k_being_Element_of_NAT_holds_(seq_^\_m)_._k_<>_0 let k be Element of NAT ; ::_thesis: (seq ^\ m) . k <> 0 seq . (k + m) < 0 by A4, A5, NAT_1:12; hence (seq ^\ m) . k <> 0 by NAT_1:def_3; ::_thesis: verum end; hence seq ^\ m is non-zero by SEQ_1:5; ::_thesis: verum end; end; end; hence ex n being Element of NAT st for m being Element of NAT st n <= m holds seq ^\ m is non-zero ; ::_thesis: verum end; theorem Th7: :: LIMFUNC1:7 for k being Element of NAT for seq being Real_Sequence holds ( ( seq ^\ k is divergent_to+infty implies seq is divergent_to+infty ) & ( seq ^\ k is divergent_to-infty implies seq is divergent_to-infty ) ) proof let k be Element of NAT ; ::_thesis: for seq being Real_Sequence holds ( ( seq ^\ k is divergent_to+infty implies seq is divergent_to+infty ) & ( seq ^\ k is divergent_to-infty implies seq is divergent_to-infty ) ) let seq be Real_Sequence; ::_thesis: ( ( seq ^\ k is divergent_to+infty implies seq is divergent_to+infty ) & ( seq ^\ k is divergent_to-infty implies seq is divergent_to-infty ) ) thus ( seq ^\ k is divergent_to+infty implies seq is divergent_to+infty ) ::_thesis: ( seq ^\ k is divergent_to-infty implies seq is divergent_to-infty ) proof assume A1: seq ^\ k is divergent_to+infty ; ::_thesis: seq is divergent_to+infty let r be Real; :: according to LIMFUNC1:def_4 ::_thesis: ex n being Element of NAT st for m being Element of NAT st n <= m holds r < seq . m consider n1 being Element of NAT such that A2: for m being Element of NAT st n1 <= m holds r < (seq ^\ k) . m by A1, Def4; take n = n1 + k; ::_thesis: for m being Element of NAT st n <= m holds r < seq . m let m be Element of NAT ; ::_thesis: ( n <= m implies r < seq . m ) assume n <= m ; ::_thesis: r < seq . m then consider n2 being Nat such that A3: m = n + n2 by NAT_1:10; reconsider n2 = n2 as Element of NAT by ORDINAL1:def_12; A4: r < (seq ^\ k) . (n1 + n2) by A2, NAT_1:12; (n1 + n2) + k = m by A3; hence r < seq . m by A4, NAT_1:def_3; ::_thesis: verum end; assume A5: seq ^\ k is divergent_to-infty ; ::_thesis: seq is divergent_to-infty let r be Real; :: according to LIMFUNC1:def_5 ::_thesis: ex n being Element of NAT st for m being Element of NAT st n <= m holds seq . m < r consider n1 being Element of NAT such that A6: for m being Element of NAT st n1 <= m holds (seq ^\ k) . m < r by A5, Def5; take n = n1 + k; ::_thesis: for m being Element of NAT st n <= m holds seq . m < r let m be Element of NAT ; ::_thesis: ( n <= m implies seq . m < r ) assume n <= m ; ::_thesis: seq . m < r then consider n2 being Nat such that A7: m = n + n2 by NAT_1:10; reconsider n2 = n2 as Element of NAT by ORDINAL1:def_12; A8: (seq ^\ k) . (n1 + n2) < r by A6, NAT_1:12; (n1 + n2) + k = m by A7; hence seq . m < r by A8, NAT_1:def_3; ::_thesis: verum end; theorem Th8: :: LIMFUNC1:8 for seq1, seq2 being Real_Sequence st seq1 is divergent_to+infty & seq2 is divergent_to+infty holds seq1 + seq2 is divergent_to+infty proof let seq1, seq2 be Real_Sequence; ::_thesis: ( seq1 is divergent_to+infty & seq2 is divergent_to+infty implies seq1 + seq2 is divergent_to+infty ) assume that A1: seq1 is divergent_to+infty and A2: seq2 is divergent_to+infty ; ::_thesis: seq1 + seq2 is divergent_to+infty let r be Real; :: according to LIMFUNC1:def_4 ::_thesis: ex n being Element of NAT st for m being Element of NAT st n <= m holds r < (seq1 + seq2) . m consider n1 being Element of NAT such that A3: for m being Element of NAT st n1 <= m holds r / 2 < seq1 . m by A1, Def4; consider n2 being Element of NAT such that A4: for m being Element of NAT st n2 <= m holds r / 2 < seq2 . m by A2, Def4; take n = max (n1,n2); ::_thesis: for m being Element of NAT st n <= m holds r < (seq1 + seq2) . m let m be Element of NAT ; ::_thesis: ( n <= m implies r < (seq1 + seq2) . m ) assume A5: n <= m ; ::_thesis: r < (seq1 + seq2) . m n2 <= n by XXREAL_0:25; then n2 <= m by A5, XXREAL_0:2; then A6: r / 2 < seq2 . m by A4; n1 <= n by XXREAL_0:25; then n1 <= m by A5, XXREAL_0:2; then r / 2 < seq1 . m by A3; then (r / 2) + (r / 2) < (seq1 . m) + (seq2 . m) by A6, XREAL_1:8; hence r < (seq1 + seq2) . m by SEQ_1:7; ::_thesis: verum end; theorem Th9: :: LIMFUNC1:9 for seq1, seq2 being Real_Sequence st seq1 is divergent_to+infty & seq2 is bounded_below holds seq1 + seq2 is divergent_to+infty proof let seq1, seq2 be Real_Sequence; ::_thesis: ( seq1 is divergent_to+infty & seq2 is bounded_below implies seq1 + seq2 is divergent_to+infty ) assume that A1: seq1 is divergent_to+infty and A2: seq2 is bounded_below ; ::_thesis: seq1 + seq2 is divergent_to+infty let r be Real; :: according to LIMFUNC1:def_4 ::_thesis: ex n being Element of NAT st for m being Element of NAT st n <= m holds r < (seq1 + seq2) . m consider M being real number such that A3: for n being Element of NAT holds M < seq2 . n by A2, SEQ_2:def_4; consider n being Element of NAT such that A4: for m being Element of NAT st n <= m holds r - M < seq1 . m by A1, Def4; take n ; ::_thesis: for m being Element of NAT st n <= m holds r < (seq1 + seq2) . m let m be Element of NAT ; ::_thesis: ( n <= m implies r < (seq1 + seq2) . m ) assume n <= m ; ::_thesis: r < (seq1 + seq2) . m then r - M < seq1 . m by A4; then (r - M) + M < (seq1 . m) + (seq2 . m) by A3, XREAL_1:8; hence r < (seq1 + seq2) . m by SEQ_1:7; ::_thesis: verum end; theorem Th10: :: LIMFUNC1:10 for seq1, seq2 being Real_Sequence st seq1 is divergent_to+infty & seq2 is divergent_to+infty holds seq1 (#) seq2 is divergent_to+infty proof let seq1, seq2 be Real_Sequence; ::_thesis: ( seq1 is divergent_to+infty & seq2 is divergent_to+infty implies seq1 (#) seq2 is divergent_to+infty ) assume that A1: seq1 is divergent_to+infty and A2: seq2 is divergent_to+infty ; ::_thesis: seq1 (#) seq2 is divergent_to+infty let r be Real; :: according to LIMFUNC1:def_4 ::_thesis: ex n being Element of NAT st for m being Element of NAT st n <= m holds r < (seq1 (#) seq2) . m consider n1 being Element of NAT such that A3: for m being Element of NAT st n1 <= m holds sqrt (abs r) < seq1 . m by A1, Def4; consider n2 being Element of NAT such that A4: for m being Element of NAT st n2 <= m holds sqrt (abs r) < seq2 . m by A2, Def4; take n = max (n1,n2); ::_thesis: for m being Element of NAT st n <= m holds r < (seq1 (#) seq2) . m let m be Element of NAT ; ::_thesis: ( n <= m implies r < (seq1 (#) seq2) . m ) assume A5: n <= m ; ::_thesis: r < (seq1 (#) seq2) . m n2 <= n by XXREAL_0:25; then n2 <= m by A5, XXREAL_0:2; then A6: sqrt (abs r) < seq2 . m by A4; n1 <= n by XXREAL_0:25; then n1 <= m by A5, XXREAL_0:2; then A7: sqrt (abs r) < seq1 . m by A3; A8: abs r >= 0 by COMPLEX1:46; then sqrt (abs r) >= 0 by SQUARE_1:def_2; then (sqrt (abs r)) ^2 < (seq1 . m) * (seq2 . m) by A7, A6, XREAL_1:96; then A9: abs r < (seq1 . m) * (seq2 . m) by A8, SQUARE_1:def_2; r <= abs r by ABSVALUE:4; then r < (seq1 . m) * (seq2 . m) by A9, XXREAL_0:2; hence r < (seq1 (#) seq2) . m by SEQ_1:8; ::_thesis: verum end; theorem Th11: :: LIMFUNC1:11 for seq1, seq2 being Real_Sequence st seq1 is divergent_to-infty & seq2 is divergent_to-infty holds seq1 + seq2 is divergent_to-infty proof let seq1, seq2 be Real_Sequence; ::_thesis: ( seq1 is divergent_to-infty & seq2 is divergent_to-infty implies seq1 + seq2 is divergent_to-infty ) assume that A1: seq1 is divergent_to-infty and A2: seq2 is divergent_to-infty ; ::_thesis: seq1 + seq2 is divergent_to-infty let r be Real; :: according to LIMFUNC1:def_5 ::_thesis: ex n being Element of NAT st for m being Element of NAT st n <= m holds (seq1 + seq2) . m < r consider n1 being Element of NAT such that A3: for m being Element of NAT st n1 <= m holds seq1 . m < r / 2 by A1, Def5; consider n2 being Element of NAT such that A4: for m being Element of NAT st n2 <= m holds seq2 . m < r / 2 by A2, Def5; take n = max (n1,n2); ::_thesis: for m being Element of NAT st n <= m holds (seq1 + seq2) . m < r let m be Element of NAT ; ::_thesis: ( n <= m implies (seq1 + seq2) . m < r ) assume A5: n <= m ; ::_thesis: (seq1 + seq2) . m < r n2 <= n by XXREAL_0:25; then n2 <= m by A5, XXREAL_0:2; then A6: seq2 . m < r / 2 by A4; n1 <= n by XXREAL_0:25; then n1 <= m by A5, XXREAL_0:2; then seq1 . m < r / 2 by A3; then (seq1 . m) + (seq2 . m) < (r / 2) + (r / 2) by A6, XREAL_1:8; hence (seq1 + seq2) . m < r by SEQ_1:7; ::_thesis: verum end; theorem Th12: :: LIMFUNC1:12 for seq1, seq2 being Real_Sequence st seq1 is divergent_to-infty & seq2 is bounded_above holds seq1 + seq2 is divergent_to-infty proof let seq1, seq2 be Real_Sequence; ::_thesis: ( seq1 is divergent_to-infty & seq2 is bounded_above implies seq1 + seq2 is divergent_to-infty ) assume that A1: seq1 is divergent_to-infty and A2: seq2 is bounded_above ; ::_thesis: seq1 + seq2 is divergent_to-infty let r be Real; :: according to LIMFUNC1:def_5 ::_thesis: ex n being Element of NAT st for m being Element of NAT st n <= m holds (seq1 + seq2) . m < r consider M being real number such that A3: for n being Element of NAT holds seq2 . n < M by A2, SEQ_2:def_3; consider n being Element of NAT such that A4: for m being Element of NAT st n <= m holds seq1 . m < r - M by A1, Def5; take n ; ::_thesis: for m being Element of NAT st n <= m holds (seq1 + seq2) . m < r let m be Element of NAT ; ::_thesis: ( n <= m implies (seq1 + seq2) . m < r ) assume n <= m ; ::_thesis: (seq1 + seq2) . m < r then seq1 . m < r - M by A4; then (seq1 . m) + (seq2 . m) < (r - M) + M by A3, XREAL_1:8; hence (seq1 + seq2) . m < r by SEQ_1:7; ::_thesis: verum end; theorem Th13: :: LIMFUNC1:13 for seq being Real_Sequence for r being Real holds ( ( seq is divergent_to+infty & r > 0 implies r (#) seq is divergent_to+infty ) & ( seq is divergent_to+infty & r < 0 implies r (#) seq is divergent_to-infty ) & ( r = 0 implies ( rng (r (#) seq) = {0} & r (#) seq is constant ) ) ) proof let seq be Real_Sequence; ::_thesis: for r being Real holds ( ( seq is divergent_to+infty & r > 0 implies r (#) seq is divergent_to+infty ) & ( seq is divergent_to+infty & r < 0 implies r (#) seq is divergent_to-infty ) & ( r = 0 implies ( rng (r (#) seq) = {0} & r (#) seq is constant ) ) ) let r be Real; ::_thesis: ( ( seq is divergent_to+infty & r > 0 implies r (#) seq is divergent_to+infty ) & ( seq is divergent_to+infty & r < 0 implies r (#) seq is divergent_to-infty ) & ( r = 0 implies ( rng (r (#) seq) = {0} & r (#) seq is constant ) ) ) thus ( seq is divergent_to+infty & r > 0 implies r (#) seq is divergent_to+infty ) ::_thesis: ( ( seq is divergent_to+infty & r < 0 implies r (#) seq is divergent_to-infty ) & ( r = 0 implies ( rng (r (#) seq) = {0} & r (#) seq is constant ) ) ) proof assume that A1: seq is divergent_to+infty and A2: r > 0 ; ::_thesis: r (#) seq is divergent_to+infty let g be Real; :: according to LIMFUNC1:def_4 ::_thesis: ex n being Element of NAT st for m being Element of NAT st n <= m holds g < (r (#) seq) . m consider n being Element of NAT such that A3: for m being Element of NAT st n <= m holds g / r < seq . m by A1, Def4; take n ; ::_thesis: for m being Element of NAT st n <= m holds g < (r (#) seq) . m let m be Element of NAT ; ::_thesis: ( n <= m implies g < (r (#) seq) . m ) assume n <= m ; ::_thesis: g < (r (#) seq) . m then g / r < seq . m by A3; then (g / r) * r < r * (seq . m) by A2, XREAL_1:68; then g < r * (seq . m) by A2, XCMPLX_1:87; hence g < (r (#) seq) . m by SEQ_1:9; ::_thesis: verum end; thus ( seq is divergent_to+infty & r < 0 implies r (#) seq is divergent_to-infty ) ::_thesis: ( r = 0 implies ( rng (r (#) seq) = {0} & r (#) seq is constant ) ) proof assume that A4: seq is divergent_to+infty and A5: r < 0 ; ::_thesis: r (#) seq is divergent_to-infty let g be Real; :: according to LIMFUNC1:def_5 ::_thesis: ex n being Element of NAT st for m being Element of NAT st n <= m holds (r (#) seq) . m < g consider n being Element of NAT such that A6: for m being Element of NAT st n <= m holds g / r < seq . m by A4, Def4; take n ; ::_thesis: for m being Element of NAT st n <= m holds (r (#) seq) . m < g let m be Element of NAT ; ::_thesis: ( n <= m implies (r (#) seq) . m < g ) assume n <= m ; ::_thesis: (r (#) seq) . m < g then g / r < seq . m by A6; then r * (seq . m) < (g / r) * r by A5, XREAL_1:69; then r * (seq . m) < g by A5, XCMPLX_1:87; hence (r (#) seq) . m < g by SEQ_1:9; ::_thesis: verum end; assume A7: r = 0 ; ::_thesis: ( rng (r (#) seq) = {0} & r (#) seq is constant ) thus rng (r (#) seq) = {0} ::_thesis: r (#) seq is constant proof let x be real number ; :: according to MEMBERED:def_15 ::_thesis: ( ( not x in rng (r (#) seq) or x in {0} ) & ( not x in {0} or x in rng (r (#) seq) ) ) thus ( x in rng (r (#) seq) implies x in {0} ) ::_thesis: ( not x in {0} or x in rng (r (#) seq) ) proof assume x in rng (r (#) seq) ; ::_thesis: x in {0} then consider n being Element of NAT such that A8: x = (r (#) seq) . n by FUNCT_2:113; x = r * (seq . n) by A8, SEQ_1:9 .= 0 by A7 ; hence x in {0} by TARSKI:def_1; ::_thesis: verum end; assume x in {0} ; ::_thesis: x in rng (r (#) seq) then A9: x = 0 by TARSKI:def_1; (r (#) seq) . 0 = r * (seq . 0) by SEQ_1:9 .= 0 by A7 ; hence x in rng (r (#) seq) by A9, VALUED_0:28; ::_thesis: verum end; hence r (#) seq is constant by FUNCT_2:111; ::_thesis: verum end; theorem Th14: :: LIMFUNC1:14 for seq being Real_Sequence for r being Real holds ( ( seq is divergent_to-infty & r > 0 implies r (#) seq is divergent_to-infty ) & ( seq is divergent_to-infty & r < 0 implies r (#) seq is divergent_to+infty ) & ( r = 0 implies ( rng (r (#) seq) = {0} & r (#) seq is constant ) ) ) proof let seq be Real_Sequence; ::_thesis: for r being Real holds ( ( seq is divergent_to-infty & r > 0 implies r (#) seq is divergent_to-infty ) & ( seq is divergent_to-infty & r < 0 implies r (#) seq is divergent_to+infty ) & ( r = 0 implies ( rng (r (#) seq) = {0} & r (#) seq is constant ) ) ) let r be Real; ::_thesis: ( ( seq is divergent_to-infty & r > 0 implies r (#) seq is divergent_to-infty ) & ( seq is divergent_to-infty & r < 0 implies r (#) seq is divergent_to+infty ) & ( r = 0 implies ( rng (r (#) seq) = {0} & r (#) seq is constant ) ) ) thus ( seq is divergent_to-infty & r > 0 implies r (#) seq is divergent_to-infty ) ::_thesis: ( ( seq is divergent_to-infty & r < 0 implies r (#) seq is divergent_to+infty ) & ( r = 0 implies ( rng (r (#) seq) = {0} & r (#) seq is constant ) ) ) proof assume that A1: seq is divergent_to-infty and A2: r > 0 ; ::_thesis: r (#) seq is divergent_to-infty let g be Real; :: according to LIMFUNC1:def_5 ::_thesis: ex n being Element of NAT st for m being Element of NAT st n <= m holds (r (#) seq) . m < g consider n being Element of NAT such that A3: for m being Element of NAT st n <= m holds seq . m < g / r by A1, Def5; take n ; ::_thesis: for m being Element of NAT st n <= m holds (r (#) seq) . m < g let m be Element of NAT ; ::_thesis: ( n <= m implies (r (#) seq) . m < g ) assume n <= m ; ::_thesis: (r (#) seq) . m < g then seq . m < g / r by A3; then r * (seq . m) < (g / r) * r by A2, XREAL_1:68; then r * (seq . m) < g by A2, XCMPLX_1:87; hence (r (#) seq) . m < g by SEQ_1:9; ::_thesis: verum end; thus ( seq is divergent_to-infty & r < 0 implies r (#) seq is divergent_to+infty ) ::_thesis: ( r = 0 implies ( rng (r (#) seq) = {0} & r (#) seq is constant ) ) proof assume that A4: seq is divergent_to-infty and A5: r < 0 ; ::_thesis: r (#) seq is divergent_to+infty let g be Real; :: according to LIMFUNC1:def_4 ::_thesis: ex n being Element of NAT st for m being Element of NAT st n <= m holds g < (r (#) seq) . m consider n being Element of NAT such that A6: for m being Element of NAT st n <= m holds seq . m < g / r by A4, Def5; take n ; ::_thesis: for m being Element of NAT st n <= m holds g < (r (#) seq) . m let m be Element of NAT ; ::_thesis: ( n <= m implies g < (r (#) seq) . m ) assume n <= m ; ::_thesis: g < (r (#) seq) . m then seq . m < g / r by A6; then (g / r) * r < r * (seq . m) by A5, XREAL_1:69; then g < r * (seq . m) by A5, XCMPLX_1:87; hence g < (r (#) seq) . m by SEQ_1:9; ::_thesis: verum end; assume A7: r = 0 ; ::_thesis: ( rng (r (#) seq) = {0} & r (#) seq is constant ) thus rng (r (#) seq) = {0} ::_thesis: r (#) seq is constant proof let x be real number ; :: according to MEMBERED:def_15 ::_thesis: ( ( not x in rng (r (#) seq) or x in {0} ) & ( not x in {0} or x in rng (r (#) seq) ) ) thus ( x in rng (r (#) seq) implies x in {0} ) ::_thesis: ( not x in {0} or x in rng (r (#) seq) ) proof assume x in rng (r (#) seq) ; ::_thesis: x in {0} then consider n being Element of NAT such that A8: x = (r (#) seq) . n by FUNCT_2:113; x = r * (seq . n) by A8, SEQ_1:9 .= 0 by A7 ; hence x in {0} by TARSKI:def_1; ::_thesis: verum end; assume x in {0} ; ::_thesis: x in rng (r (#) seq) then A9: x = 0 by TARSKI:def_1; (r (#) seq) . 0 = r * (seq . 0) by SEQ_1:9 .= 0 by A7 ; hence x in rng (r (#) seq) by A9, VALUED_0:28; ::_thesis: verum end; hence r (#) seq is constant by FUNCT_2:111; ::_thesis: verum end; theorem :: LIMFUNC1:15 for seq being Real_Sequence holds ( ( seq is divergent_to+infty implies - seq is divergent_to-infty ) & ( seq is divergent_to-infty implies - seq is divergent_to+infty ) ) proof let seq be Real_Sequence; ::_thesis: ( ( seq is divergent_to+infty implies - seq is divergent_to-infty ) & ( seq is divergent_to-infty implies - seq is divergent_to+infty ) ) A1: (- 1) (#) seq = - seq ; hence ( seq is divergent_to+infty implies - seq is divergent_to-infty ) by Th13; ::_thesis: ( seq is divergent_to-infty implies - seq is divergent_to+infty ) assume seq is divergent_to-infty ; ::_thesis: - seq is divergent_to+infty hence - seq is divergent_to+infty by A1, Th14; ::_thesis: verum end; theorem :: LIMFUNC1:16 for seq, seq1 being Real_Sequence st seq is bounded_below & seq1 is divergent_to-infty holds seq - seq1 is divergent_to+infty proof let seq, seq1 be Real_Sequence; ::_thesis: ( seq is bounded_below & seq1 is divergent_to-infty implies seq - seq1 is divergent_to+infty ) assume that A1: seq is bounded_below and A2: seq1 is divergent_to-infty ; ::_thesis: seq - seq1 is divergent_to+infty (- 1) (#) seq1 is divergent_to+infty by A2, Th14; hence seq - seq1 is divergent_to+infty by A1, Th9; ::_thesis: verum end; theorem :: LIMFUNC1:17 for seq, seq1 being Real_Sequence st seq is bounded_above & seq1 is divergent_to+infty holds seq - seq1 is divergent_to-infty proof let seq, seq1 be Real_Sequence; ::_thesis: ( seq is bounded_above & seq1 is divergent_to+infty implies seq - seq1 is divergent_to-infty ) assume that A1: seq is bounded_above and A2: seq1 is divergent_to+infty ; ::_thesis: seq - seq1 is divergent_to-infty (- 1) (#) seq1 is divergent_to-infty by A2, Th13; hence seq - seq1 is divergent_to-infty by A1, Th12; ::_thesis: verum end; theorem :: LIMFUNC1:18 for seq, seq1 being Real_Sequence st seq is divergent_to+infty & seq1 is convergent holds seq + seq1 is divergent_to+infty by Th9; theorem :: LIMFUNC1:19 for seq, seq1 being Real_Sequence st seq is divergent_to-infty & seq1 is convergent holds seq + seq1 is divergent_to-infty by Th12; theorem Th20: :: LIMFUNC1:20 for seq being Real_Sequence st ( for n being Element of NAT holds seq . n = n ) holds seq is divergent_to+infty proof let seq be Real_Sequence; ::_thesis: ( ( for n being Element of NAT holds seq . n = n ) implies seq is divergent_to+infty ) assume A1: for n being Element of NAT holds seq . n = n ; ::_thesis: seq is divergent_to+infty let r be Real; :: according to LIMFUNC1:def_4 ::_thesis: ex n being Element of NAT st for m being Element of NAT st n <= m holds r < seq . m consider n being Element of NAT such that A2: r < n by SEQ_4:3; take n ; ::_thesis: for m being Element of NAT st n <= m holds r < seq . m let m be Element of NAT ; ::_thesis: ( n <= m implies r < seq . m ) assume n <= m ; ::_thesis: r < seq . m then r < m by A2, XXREAL_0:2; hence r < seq . m by A1; ::_thesis: verum end; set s1 = incl NAT; Lm4: for n being Element of NAT holds (incl NAT) . n = n by FUNCT_1:18; theorem Th21: :: LIMFUNC1:21 for seq being Real_Sequence st ( for n being Element of NAT holds seq . n = - n ) holds seq is divergent_to-infty proof let seq be Real_Sequence; ::_thesis: ( ( for n being Element of NAT holds seq . n = - n ) implies seq is divergent_to-infty ) assume A1: for n being Element of NAT holds seq . n = - n ; ::_thesis: seq is divergent_to-infty A2: now__::_thesis:_for_n_being_Element_of_NAT_holds_(-_(incl_NAT))_._n_=_seq_._n let n be Element of NAT ; ::_thesis: (- (incl NAT)) . n = seq . n thus (- (incl NAT)) . n = - ((incl NAT) . n) by SEQ_1:10 .= - n by FUNCT_1:18 .= seq . n by A1 ; ::_thesis: verum end; incl NAT is divergent_to+infty by Lm4, Th20; then (- 1) (#) (incl NAT) is divergent_to-infty by Th13; hence seq is divergent_to-infty by A2, FUNCT_2:63; ::_thesis: verum end; theorem Th22: :: LIMFUNC1:22 for seq1, seq2 being Real_Sequence st seq1 is divergent_to+infty & ex r being Real st ( r > 0 & ( for n being Element of NAT holds seq2 . n >= r ) ) holds seq1 (#) seq2 is divergent_to+infty proof let seq1, seq2 be Real_Sequence; ::_thesis: ( seq1 is divergent_to+infty & ex r being Real st ( r > 0 & ( for n being Element of NAT holds seq2 . n >= r ) ) implies seq1 (#) seq2 is divergent_to+infty ) assume that A1: seq1 is divergent_to+infty and A2: ex r being Real st ( r > 0 & ( for n being Element of NAT holds seq2 . n >= r ) ) ; ::_thesis: seq1 (#) seq2 is divergent_to+infty consider M being Real such that A3: M > 0 and A4: for n being Element of NAT holds seq2 . n >= M by A2; let r be Real; :: according to LIMFUNC1:def_4 ::_thesis: ex n being Element of NAT st for m being Element of NAT st n <= m holds r < (seq1 (#) seq2) . m A5: 0 <= abs r by COMPLEX1:46; consider n being Element of NAT such that A6: for m being Element of NAT st n <= m holds (abs r) / M < seq1 . m by A1, Def4; take n ; ::_thesis: for m being Element of NAT st n <= m holds r < (seq1 (#) seq2) . m let m be Element of NAT ; ::_thesis: ( n <= m implies r < (seq1 (#) seq2) . m ) assume n <= m ; ::_thesis: r < (seq1 (#) seq2) . m then (abs r) / M < seq1 . m by A6; then ((abs r) / M) * M < (seq1 . m) * (seq2 . m) by A3, A4, A5, XREAL_1:97; then A7: abs r < (seq1 . m) * (seq2 . m) by A3, XCMPLX_1:87; r <= abs r by ABSVALUE:4; then r < (seq1 . m) * (seq2 . m) by A7, XXREAL_0:2; hence r < (seq1 (#) seq2) . m by SEQ_1:8; ::_thesis: verum end; theorem :: LIMFUNC1:23 for seq1, seq2 being Real_Sequence st seq1 is divergent_to-infty & ex r being Real st ( 0 < r & ( for n being Element of NAT holds seq2 . n >= r ) ) holds seq1 (#) seq2 is divergent_to-infty proof let seq1, seq2 be Real_Sequence; ::_thesis: ( seq1 is divergent_to-infty & ex r being Real st ( 0 < r & ( for n being Element of NAT holds seq2 . n >= r ) ) implies seq1 (#) seq2 is divergent_to-infty ) assume that A1: seq1 is divergent_to-infty and A2: ex r being Real st ( 0 < r & ( for n being Element of NAT holds seq2 . n >= r ) ) ; ::_thesis: seq1 (#) seq2 is divergent_to-infty (- 1) (#) seq1 is divergent_to+infty by A1, Th14; then A3: ((- 1) (#) seq1) (#) seq2 is divergent_to+infty by A2, Th22; (- 1) (#) (((- 1) (#) seq1) (#) seq2) = (- 1) (#) ((- 1) (#) (seq1 (#) seq2)) by SEQ_1:18 .= ((- 1) * (- 1)) (#) (seq1 (#) seq2) by SEQ_1:23 .= seq1 (#) seq2 by SEQ_1:27 ; hence seq1 (#) seq2 is divergent_to-infty by A3, Th13; ::_thesis: verum end; theorem Th24: :: LIMFUNC1:24 for seq1, seq2 being Real_Sequence st seq1 is divergent_to-infty & seq2 is divergent_to-infty holds seq1 (#) seq2 is divergent_to+infty proof let seq1, seq2 be Real_Sequence; ::_thesis: ( seq1 is divergent_to-infty & seq2 is divergent_to-infty implies seq1 (#) seq2 is divergent_to+infty ) assume ( seq1 is divergent_to-infty & seq2 is divergent_to-infty ) ; ::_thesis: seq1 (#) seq2 is divergent_to+infty then A1: ( (- 1) (#) seq1 is divergent_to+infty & (- 1) (#) seq2 is divergent_to+infty ) by Th14; ((- 1) (#) seq1) (#) ((- 1) (#) seq2) = (- 1) (#) (seq1 (#) ((- 1) (#) seq2)) by SEQ_1:18 .= (- 1) (#) ((- 1) (#) (seq1 (#) seq2)) by SEQ_1:19 .= ((- 1) * (- 1)) (#) (seq1 (#) seq2) by SEQ_1:23 .= seq1 (#) seq2 by SEQ_1:27 ; hence seq1 (#) seq2 is divergent_to+infty by A1, Th10; ::_thesis: verum end; theorem Th25: :: LIMFUNC1:25 for seq being Real_Sequence st ( seq is divergent_to+infty or seq is divergent_to-infty ) holds abs seq is divergent_to+infty proof let seq be Real_Sequence; ::_thesis: ( ( seq is divergent_to+infty or seq is divergent_to-infty ) implies abs seq is divergent_to+infty ) assume A1: ( seq is divergent_to+infty or seq is divergent_to-infty ) ; ::_thesis: abs seq is divergent_to+infty let r be Real; :: according to LIMFUNC1:def_4 ::_thesis: ex n being Element of NAT st for m being Element of NAT st n <= m holds r < (abs seq) . m now__::_thesis:_ex_n_being_Element_of_NAT_st_ for_m_being_Element_of_NAT_st_n_<=_m_holds_ r_<_(abs_seq)_._m percases ( seq is divergent_to+infty or seq is divergent_to-infty ) by A1; suppose seq is divergent_to+infty ; ::_thesis: ex n being Element of NAT st for m being Element of NAT st n <= m holds r < (abs seq) . m then consider n being Element of NAT such that A2: for m being Element of NAT st n <= m holds abs r < seq . m by Def4; take n = n; ::_thesis: for m being Element of NAT st n <= m holds r < (abs seq) . m let m be Element of NAT ; ::_thesis: ( n <= m implies r < (abs seq) . m ) assume n <= m ; ::_thesis: r < (abs seq) . m then ( r <= abs r & abs r < seq . m ) by A2, ABSVALUE:4; then A3: r < seq . m by XXREAL_0:2; seq . m <= abs (seq . m) by ABSVALUE:4; then seq . m <= (abs seq) . m by SEQ_1:12; hence r < (abs seq) . m by A3, XXREAL_0:2; ::_thesis: verum end; suppose seq is divergent_to-infty ; ::_thesis: ex n being Element of NAT st for m being Element of NAT st n <= m holds r < (abs seq) . m then consider n being Element of NAT such that A4: for m being Element of NAT st n <= m holds seq . m < - r by Def5; take n = n; ::_thesis: for m being Element of NAT st n <= m holds r < (abs seq) . m let m be Element of NAT ; ::_thesis: ( n <= m implies r < (abs seq) . m ) - (abs (seq . m)) <= seq . m by ABSVALUE:4; then A5: - ((abs seq) . m) <= seq . m by SEQ_1:12; assume n <= m ; ::_thesis: r < (abs seq) . m then seq . m < - r by A4; then - ((abs seq) . m) < - r by A5, XXREAL_0:2; hence r < (abs seq) . m by XREAL_1:24; ::_thesis: verum end; end; end; hence ex n being Element of NAT st for m being Element of NAT st n <= m holds r < (abs seq) . m ; ::_thesis: verum end; theorem Th26: :: LIMFUNC1:26 for seq, seq1 being Real_Sequence st seq is divergent_to+infty & seq1 is subsequence of seq holds seq1 is divergent_to+infty proof let seq, seq1 be Real_Sequence; ::_thesis: ( seq is divergent_to+infty & seq1 is subsequence of seq implies seq1 is divergent_to+infty ) assume that A1: seq is divergent_to+infty and A2: seq1 is subsequence of seq ; ::_thesis: seq1 is divergent_to+infty consider Ns being V41() sequence of NAT such that A3: seq1 = seq * Ns by A2, VALUED_0:def_17; let r be Real; :: according to LIMFUNC1:def_4 ::_thesis: ex n being Element of NAT st for m being Element of NAT st n <= m holds r < seq1 . m consider n being Element of NAT such that A4: for m being Element of NAT st n <= m holds r < seq . m by A1, Def4; take n ; ::_thesis: for m being Element of NAT st n <= m holds r < seq1 . m let m be Element of NAT ; ::_thesis: ( n <= m implies r < seq1 . m ) assume A5: n <= m ; ::_thesis: r < seq1 . m m <= Ns . m by SEQM_3:14; then n <= Ns . m by A5, XXREAL_0:2; then r < seq . (Ns . m) by A4; hence r < seq1 . m by A3, FUNCT_2:15; ::_thesis: verum end; theorem Th27: :: LIMFUNC1:27 for seq, seq1 being Real_Sequence st seq is divergent_to-infty & seq1 is subsequence of seq holds seq1 is divergent_to-infty proof let seq, seq1 be Real_Sequence; ::_thesis: ( seq is divergent_to-infty & seq1 is subsequence of seq implies seq1 is divergent_to-infty ) assume that A1: seq is divergent_to-infty and A2: seq1 is subsequence of seq ; ::_thesis: seq1 is divergent_to-infty consider Ns being V41() sequence of NAT such that A3: seq1 = seq * Ns by A2, VALUED_0:def_17; let r be Real; :: according to LIMFUNC1:def_5 ::_thesis: ex n being Element of NAT st for m being Element of NAT st n <= m holds seq1 . m < r consider n being Element of NAT such that A4: for m being Element of NAT st n <= m holds seq . m < r by A1, Def5; take n ; ::_thesis: for m being Element of NAT st n <= m holds seq1 . m < r let m be Element of NAT ; ::_thesis: ( n <= m implies seq1 . m < r ) assume A5: n <= m ; ::_thesis: seq1 . m < r m <= Ns . m by SEQM_3:14; then n <= Ns . m by A5, XXREAL_0:2; then seq . (Ns . m) < r by A4; hence seq1 . m < r by A3, FUNCT_2:15; ::_thesis: verum end; theorem :: LIMFUNC1:28 for seq1, seq2 being Real_Sequence st seq1 is divergent_to+infty & seq2 is convergent & 0 < lim seq2 holds seq1 (#) seq2 is divergent_to+infty proof let seq1, seq2 be Real_Sequence; ::_thesis: ( seq1 is divergent_to+infty & seq2 is convergent & 0 < lim seq2 implies seq1 (#) seq2 is divergent_to+infty ) assume that A1: seq1 is divergent_to+infty and A2: seq2 is convergent and A3: 0 < lim seq2 ; ::_thesis: seq1 (#) seq2 is divergent_to+infty consider n1 being Element of NAT such that A4: for m being Element of NAT st n1 <= m holds (lim seq2) / 2 < seq2 . m by A2, A3, Th5; now__::_thesis:_(_0_<_(lim_seq2)_/_2_&_(_for_n_being_Element_of_NAT_holds_(lim_seq2)_/_2_<=_(seq2_^\_n1)_._n_)_) thus 0 < (lim seq2) / 2 by A3, XREAL_1:215; ::_thesis: for n being Element of NAT holds (lim seq2) / 2 <= (seq2 ^\ n1) . n let n be Element of NAT ; ::_thesis: (lim seq2) / 2 <= (seq2 ^\ n1) . n (lim seq2) / 2 < seq2 . (n + n1) by A4, NAT_1:12; hence (lim seq2) / 2 <= (seq2 ^\ n1) . n by NAT_1:def_3; ::_thesis: verum end; then (seq1 ^\ n1) (#) (seq2 ^\ n1) is divergent_to+infty by A1, Th22, Th26; then (seq1 (#) seq2) ^\ n1 is divergent_to+infty by SEQM_3:19; hence seq1 (#) seq2 is divergent_to+infty by Th7; ::_thesis: verum end; theorem Th29: :: LIMFUNC1:29 for seq being Real_Sequence st seq is non-decreasing & not seq is bounded_above holds seq is divergent_to+infty proof let seq be Real_Sequence; ::_thesis: ( seq is non-decreasing & not seq is bounded_above implies seq is divergent_to+infty ) assume that A1: seq is non-decreasing and A2: not seq is bounded_above ; ::_thesis: seq is divergent_to+infty let r be Real; :: according to LIMFUNC1:def_4 ::_thesis: ex n being Element of NAT st for m being Element of NAT st n <= m holds r < seq . m consider n being Element of NAT such that A3: r + 1 <= seq . n by A2, SEQ_2:def_3; take n ; ::_thesis: for m being Element of NAT st n <= m holds r < seq . m let m be Element of NAT ; ::_thesis: ( n <= m implies r < seq . m ) assume n <= m ; ::_thesis: r < seq . m then seq . n <= seq . m by A1, SEQM_3:6; then r + 1 <= seq . m by A3, XXREAL_0:2; hence r < seq . m by Lm1; ::_thesis: verum end; theorem Th30: :: LIMFUNC1:30 for seq being Real_Sequence st seq is non-increasing & not seq is bounded_below holds seq is divergent_to-infty proof let seq be Real_Sequence; ::_thesis: ( seq is non-increasing & not seq is bounded_below implies seq is divergent_to-infty ) assume that A1: seq is non-increasing and A2: not seq is bounded_below ; ::_thesis: seq is divergent_to-infty let r be Real; :: according to LIMFUNC1:def_5 ::_thesis: ex n being Element of NAT st for m being Element of NAT st n <= m holds seq . m < r consider n being Element of NAT such that A3: seq . n <= r - 1 by A2, SEQ_2:def_4; take n ; ::_thesis: for m being Element of NAT st n <= m holds seq . m < r let m be Element of NAT ; ::_thesis: ( n <= m implies seq . m < r ) assume n <= m ; ::_thesis: seq . m < r then seq . m <= seq . n by A1, SEQM_3:8; then seq . m <= r - 1 by A3, XXREAL_0:2; hence seq . m < r by Lm1; ::_thesis: verum end; theorem :: LIMFUNC1:31 for seq being Real_Sequence st seq is increasing & not seq is bounded_above holds seq is divergent_to+infty by Th29; theorem :: LIMFUNC1:32 for seq being Real_Sequence st seq is decreasing & not seq is bounded_below holds seq is divergent_to-infty by Th30; theorem :: LIMFUNC1:33 for seq being Real_Sequence holds ( not seq is monotone or seq is convergent or seq is divergent_to+infty or seq is divergent_to-infty ) proof let seq be Real_Sequence; ::_thesis: ( not seq is monotone or seq is convergent or seq is divergent_to+infty or seq is divergent_to-infty ) assume A1: seq is monotone ; ::_thesis: ( seq is convergent or seq is divergent_to+infty or seq is divergent_to-infty ) now__::_thesis:_(_seq_is_convergent_or_seq_is_divergent_to+infty_or_seq_is_divergent_to-infty_) percases ( seq is non-decreasing or seq is non-increasing ) by A1, SEQM_3:def_5; supposeA2: seq is non-decreasing ; ::_thesis: ( seq is convergent or seq is divergent_to+infty or seq is divergent_to-infty ) now__::_thesis:_(_seq_is_convergent_or_seq_is_divergent_to+infty_or_seq_is_divergent_to-infty_) percases ( seq is bounded_above or not seq is bounded_above ) ; suppose seq is bounded_above ; ::_thesis: ( seq is convergent or seq is divergent_to+infty or seq is divergent_to-infty ) hence ( seq is convergent or seq is divergent_to+infty or seq is divergent_to-infty ) by A2; ::_thesis: verum end; suppose not seq is bounded_above ; ::_thesis: ( seq is convergent or seq is divergent_to+infty or seq is divergent_to-infty ) hence ( seq is convergent or seq is divergent_to+infty or seq is divergent_to-infty ) by A2, Th29; ::_thesis: verum end; end; end; hence ( seq is convergent or seq is divergent_to+infty or seq is divergent_to-infty ) ; ::_thesis: verum end; supposeA3: seq is non-increasing ; ::_thesis: ( seq is convergent or seq is divergent_to+infty or seq is divergent_to-infty ) now__::_thesis:_(_seq_is_convergent_or_seq_is_divergent_to+infty_or_seq_is_divergent_to-infty_) percases ( seq is bounded_below or not seq is bounded_below ) ; suppose seq is bounded_below ; ::_thesis: ( seq is convergent or seq is divergent_to+infty or seq is divergent_to-infty ) hence ( seq is convergent or seq is divergent_to+infty or seq is divergent_to-infty ) by A3; ::_thesis: verum end; suppose not seq is bounded_below ; ::_thesis: ( seq is convergent or seq is divergent_to+infty or seq is divergent_to-infty ) hence ( seq is convergent or seq is divergent_to+infty or seq is divergent_to-infty ) by A3, Th30; ::_thesis: verum end; end; end; hence ( seq is convergent or seq is divergent_to+infty or seq is divergent_to-infty ) ; ::_thesis: verum end; end; end; hence ( seq is convergent or seq is divergent_to+infty or seq is divergent_to-infty ) ; ::_thesis: verum end; theorem Th34: :: LIMFUNC1:34 for seq being Real_Sequence st ( seq is divergent_to+infty or seq is divergent_to-infty ) holds ( seq " is convergent & lim (seq ") = 0 ) proof let seq be Real_Sequence; ::_thesis: ( ( seq is divergent_to+infty or seq is divergent_to-infty ) implies ( seq " is convergent & lim (seq ") = 0 ) ) assume A1: ( seq is divergent_to+infty or seq is divergent_to-infty ) ; ::_thesis: ( seq " is convergent & lim (seq ") = 0 ) now__::_thesis:_(_seq_"_is_convergent_&_seq_"_is_convergent_&_lim_(seq_")_=_0_) percases ( seq is divergent_to+infty or seq is divergent_to-infty ) by A1; supposeA2: seq is divergent_to+infty ; ::_thesis: ( seq " is convergent & seq " is convergent & lim (seq ") = 0 ) A3: now__::_thesis:_for_r_being_real_number_st_0_<_r_holds_ ex_n_being_Element_of_NAT_st_ for_m_being_Element_of_NAT_st_n_<=_m_holds_ abs_(((seq_")_._m)_-_0)_<_r let r be real number ; ::_thesis: ( 0 < r implies ex n being Element of NAT st for m being Element of NAT st n <= m holds abs (((seq ") . m) - 0) < r ) assume A4: 0 < r ; ::_thesis: ex n being Element of NAT st for m being Element of NAT st n <= m holds abs (((seq ") . m) - 0) < r r " is Real by XREAL_0:def_1; then consider n being Element of NAT such that A5: for m being Element of NAT st n <= m holds r " < seq . m by A2, Def4; take n = n; ::_thesis: for m being Element of NAT st n <= m holds abs (((seq ") . m) - 0) < r let m be Element of NAT ; ::_thesis: ( n <= m implies abs (((seq ") . m) - 0) < r ) assume n <= m ; ::_thesis: abs (((seq ") . m) - 0) < r then A6: r " < seq . m by A5; then 1 / (seq . m) < 1 / (r ") by A4, XREAL_1:76; then A7: 1 / (seq . m) < r by XCMPLX_1:216; A8: ( 1 / (seq . m) = (seq . m) " & (seq . m) " = (seq ") . m ) by VALUED_1:10, XCMPLX_1:215; 0 < r " by A4; hence abs (((seq ") . m) - 0) < r by A6, A7, A8, ABSVALUE:def_1; ::_thesis: verum end; hence seq " is convergent by SEQ_2:def_6; ::_thesis: ( seq " is convergent & lim (seq ") = 0 ) hence ( seq " is convergent & lim (seq ") = 0 ) by A3, SEQ_2:def_7; ::_thesis: verum end; supposeA9: seq is divergent_to-infty ; ::_thesis: ( seq " is convergent & seq " is convergent & lim (seq ") = 0 ) A10: now__::_thesis:_for_r_being_real_number_st_0_<_r_holds_ ex_n_being_Element_of_NAT_st_ for_m_being_Element_of_NAT_st_n_<=_m_holds_ abs_(((seq_")_._m)_-_0)_<_r let r be real number ; ::_thesis: ( 0 < r implies ex n being Element of NAT st for m being Element of NAT st n <= m holds abs (((seq ") . m) - 0) < r ) assume A11: 0 < r ; ::_thesis: ex n being Element of NAT st for m being Element of NAT st n <= m holds abs (((seq ") . m) - 0) < r A12: - (r ") < - 0 by A11, XREAL_1:24; - (r ") is Real by XREAL_0:def_1; then consider n being Element of NAT such that A13: for m being Element of NAT st n <= m holds seq . m < - (r ") by A9, Def5; take n = n; ::_thesis: for m being Element of NAT st n <= m holds abs (((seq ") . m) - 0) < r let m be Element of NAT ; ::_thesis: ( n <= m implies abs (((seq ") . m) - 0) < r ) assume A14: n <= m ; ::_thesis: abs (((seq ") . m) - 0) < r then seq . m < - (r ") by A13; then 1 / (- (r ")) < 1 / (seq . m) by A12, XREAL_1:99; then ((- 1) * (r ")) " < 1 / (seq . m) by XCMPLX_1:215; then A15: ((- 1) ") * ((r ") ") < 1 / (seq . m) by XCMPLX_1:204; seq . m < - 0 by A11, A13, A14; then 1 / (seq . m) < 0 / (seq . m) by XREAL_1:75; then abs (1 / (seq . m)) = - (1 / (seq . m)) by ABSVALUE:def_1; then - (1 * r) < - (abs (1 / (seq . m))) by A15; then abs (1 / (seq . m)) < r by XREAL_1:24; then abs ((seq . m) ") < r by XCMPLX_1:215; hence abs (((seq ") . m) - 0) < r by VALUED_1:10; ::_thesis: verum end; hence seq " is convergent by SEQ_2:def_6; ::_thesis: ( seq " is convergent & lim (seq ") = 0 ) hence ( seq " is convergent & lim (seq ") = 0 ) by A10, SEQ_2:def_7; ::_thesis: verum end; end; end; hence ( seq " is convergent & lim (seq ") = 0 ) ; ::_thesis: verum end; theorem Th35: :: LIMFUNC1:35 for seq being Real_Sequence st seq is convergent & lim seq = 0 & ex k being Element of NAT st for n being Element of NAT st k <= n holds 0 < seq . n holds seq " is divergent_to+infty proof let seq be Real_Sequence; ::_thesis: ( seq is convergent & lim seq = 0 & ex k being Element of NAT st for n being Element of NAT st k <= n holds 0 < seq . n implies seq " is divergent_to+infty ) assume A1: ( seq is convergent & lim seq = 0 ) ; ::_thesis: ( for k being Element of NAT ex n being Element of NAT st ( k <= n & not 0 < seq . n ) or seq " is divergent_to+infty ) given k being Element of NAT such that A2: for n being Element of NAT st k <= n holds 0 < seq . n ; ::_thesis: seq " is divergent_to+infty let r be Real; :: according to LIMFUNC1:def_4 ::_thesis: ex n being Element of NAT st for m being Element of NAT st n <= m holds r < (seq ") . m set l = (abs r) + 1; 0 <= abs r by COMPLEX1:46; then consider o being Element of NAT such that A3: for n being Element of NAT st o <= n holds abs ((seq . n) - 0) < ((abs r) + 1) " by A1, SEQ_2:def_7; take m = max (k,o); ::_thesis: for m being Element of NAT st m <= m holds r < (seq ") . m let n be Element of NAT ; ::_thesis: ( m <= n implies r < (seq ") . n ) assume A4: m <= n ; ::_thesis: r < (seq ") . n k <= m by XXREAL_0:25; then k <= n by A4, XXREAL_0:2; then A5: 0 < seq . n by A2; o <= m by XXREAL_0:25; then o <= n by A4, XXREAL_0:2; then abs ((seq . n) - 0) < ((abs r) + 1) " by A3; then seq . n < ((abs r) + 1) " by A5, ABSVALUE:def_1; then 1 / (((abs r) + 1) ") < 1 / (seq . n) by A5, XREAL_1:76; then A6: (abs r) + 1 < 1 / (seq . n) by XCMPLX_1:216; r <= abs r by ABSVALUE:4; then r < (abs r) + 1 by Lm1; then r < 1 / (seq . n) by A6, XXREAL_0:2; then r < (seq . n) " by XCMPLX_1:215; hence r < (seq ") . n by VALUED_1:10; ::_thesis: verum end; theorem Th36: :: LIMFUNC1:36 for seq being Real_Sequence st seq is convergent & lim seq = 0 & ex k being Element of NAT st for n being Element of NAT st k <= n holds seq . n < 0 holds seq " is divergent_to-infty proof let seq be Real_Sequence; ::_thesis: ( seq is convergent & lim seq = 0 & ex k being Element of NAT st for n being Element of NAT st k <= n holds seq . n < 0 implies seq " is divergent_to-infty ) assume A1: ( seq is convergent & lim seq = 0 ) ; ::_thesis: ( for k being Element of NAT ex n being Element of NAT st ( k <= n & not seq . n < 0 ) or seq " is divergent_to-infty ) given k being Element of NAT such that A2: for n being Element of NAT st k <= n holds seq . n < 0 ; ::_thesis: seq " is divergent_to-infty let r be Real; :: according to LIMFUNC1:def_5 ::_thesis: ex n being Element of NAT st for m being Element of NAT st n <= m holds (seq ") . m < r set l = (abs r) + 1; 0 <= abs r by COMPLEX1:46; then consider o being Element of NAT such that A3: for n being Element of NAT st o <= n holds abs ((seq . n) - 0) < ((abs r) + 1) " by A1, SEQ_2:def_7; take m = max (k,o); ::_thesis: for m being Element of NAT st m <= m holds (seq ") . m < r let n be Element of NAT ; ::_thesis: ( m <= n implies (seq ") . n < r ) assume A4: m <= n ; ::_thesis: (seq ") . n < r k <= m by XXREAL_0:25; then k <= n by A4, XXREAL_0:2; then A5: seq . n < 0 by A2; then A6: 0 < - (seq . n) by XREAL_1:58; o <= m by XXREAL_0:25; then o <= n by A4, XXREAL_0:2; then abs ((seq . n) - 0) < ((abs r) + 1) " by A3; then - (seq . n) < ((abs r) + 1) " by A5, ABSVALUE:def_1; then 1 / (((abs r) + 1) ") < 1 / (- (seq . n)) by A6, XREAL_1:76; then (abs r) + 1 < 1 / (- (seq . n)) by XCMPLX_1:216; then (abs r) + 1 < (- (seq . n)) " by XCMPLX_1:215; then (abs r) + 1 < - ((seq . n) ") by XCMPLX_1:222; then A7: - (- ((seq . n) ")) < - ((abs r) + 1) by XREAL_1:24; - (abs r) <= r by ABSVALUE:4; then (- (abs r)) - 1 < r by Lm1; then (seq . n) " < r by A7, XXREAL_0:2; hence (seq ") . n < r by VALUED_1:10; ::_thesis: verum end; theorem Th37: :: LIMFUNC1:37 for seq being Real_Sequence st seq is non-zero & seq is convergent & lim seq = 0 & seq is non-decreasing holds seq " is divergent_to-infty proof let seq be Real_Sequence; ::_thesis: ( seq is non-zero & seq is convergent & lim seq = 0 & seq is non-decreasing implies seq " is divergent_to-infty ) assume that A1: seq is non-zero and A2: ( seq is convergent & lim seq = 0 ) and A3: seq is non-decreasing ; ::_thesis: seq " is divergent_to-infty for n being Element of NAT st 0 <= n holds seq . n < 0 by A1, A2, A3, Th2; hence seq " is divergent_to-infty by A2, Th36; ::_thesis: verum end; theorem Th38: :: LIMFUNC1:38 for seq being Real_Sequence st seq is non-zero & seq is convergent & lim seq = 0 & seq is non-increasing holds seq " is divergent_to+infty proof let seq be Real_Sequence; ::_thesis: ( seq is non-zero & seq is convergent & lim seq = 0 & seq is non-increasing implies seq " is divergent_to+infty ) assume that A1: seq is non-zero and A2: ( seq is convergent & lim seq = 0 ) and A3: seq is non-increasing ; ::_thesis: seq " is divergent_to+infty for n being Element of NAT st 0 <= n holds 0 < seq . n by A1, A2, A3, Th3; hence seq " is divergent_to+infty by A2, Th35; ::_thesis: verum end; theorem :: LIMFUNC1:39 for seq being Real_Sequence st seq is non-zero & seq is convergent & lim seq = 0 & seq is increasing holds seq " is divergent_to-infty by Th37; theorem :: LIMFUNC1:40 for seq being Real_Sequence st seq is non-zero & seq is convergent & lim seq = 0 & seq is decreasing holds seq " is divergent_to+infty by Th38; theorem :: LIMFUNC1:41 for seq1, seq2 being Real_Sequence st seq1 is bounded & ( seq2 is divergent_to+infty or seq2 is divergent_to-infty ) holds ( seq1 /" seq2 is convergent & lim (seq1 /" seq2) = 0 ) proof let seq1, seq2 be Real_Sequence; ::_thesis: ( seq1 is bounded & ( seq2 is divergent_to+infty or seq2 is divergent_to-infty ) implies ( seq1 /" seq2 is convergent & lim (seq1 /" seq2) = 0 ) ) assume that A1: seq1 is bounded and A2: ( seq2 is divergent_to+infty or seq2 is divergent_to-infty ) ; ::_thesis: ( seq1 /" seq2 is convergent & lim (seq1 /" seq2) = 0 ) ( seq2 " is convergent & lim (seq2 ") = 0 ) by A2, Th34; hence ( seq1 /" seq2 is convergent & lim (seq1 /" seq2) = 0 ) by A1, SEQ_2:25, SEQ_2:26; ::_thesis: verum end; theorem Th42: :: LIMFUNC1:42 for seq, seq1 being Real_Sequence st seq is divergent_to+infty & ( for n being Element of NAT holds seq . n <= seq1 . n ) holds seq1 is divergent_to+infty proof let seq, seq1 be Real_Sequence; ::_thesis: ( seq is divergent_to+infty & ( for n being Element of NAT holds seq . n <= seq1 . n ) implies seq1 is divergent_to+infty ) assume that A1: seq is divergent_to+infty and A2: for n being Element of NAT holds seq . n <= seq1 . n ; ::_thesis: seq1 is divergent_to+infty let r be Real; :: according to LIMFUNC1:def_4 ::_thesis: ex n being Element of NAT st for m being Element of NAT st n <= m holds r < seq1 . m consider n being Element of NAT such that A3: for m being Element of NAT st n <= m holds r < seq . m by A1, Def4; take n ; ::_thesis: for m being Element of NAT st n <= m holds r < seq1 . m let m be Element of NAT ; ::_thesis: ( n <= m implies r < seq1 . m ) assume n <= m ; ::_thesis: r < seq1 . m then A4: r < seq . m by A3; seq . m <= seq1 . m by A2; hence r < seq1 . m by A4, XXREAL_0:2; ::_thesis: verum end; theorem Th43: :: LIMFUNC1:43 for seq, seq1 being Real_Sequence st seq is divergent_to-infty & ( for n being Element of NAT holds seq1 . n <= seq . n ) holds seq1 is divergent_to-infty proof let seq, seq1 be Real_Sequence; ::_thesis: ( seq is divergent_to-infty & ( for n being Element of NAT holds seq1 . n <= seq . n ) implies seq1 is divergent_to-infty ) assume that A1: seq is divergent_to-infty and A2: for n being Element of NAT holds seq1 . n <= seq . n ; ::_thesis: seq1 is divergent_to-infty let r be Real; :: according to LIMFUNC1:def_5 ::_thesis: ex n being Element of NAT st for m being Element of NAT st n <= m holds seq1 . m < r consider n being Element of NAT such that A3: for m being Element of NAT st n <= m holds seq . m < r by A1, Def5; take n ; ::_thesis: for m being Element of NAT st n <= m holds seq1 . m < r let m be Element of NAT ; ::_thesis: ( n <= m implies seq1 . m < r ) assume n <= m ; ::_thesis: seq1 . m < r then A4: seq . m < r by A3; seq1 . m <= seq . m by A2; hence seq1 . m < r by A4, XXREAL_0:2; ::_thesis: verum end; definition let f be PartFunc of REAL,REAL; attrf is convergent_in+infty means :Def6: :: LIMFUNC1:def 6 ( ( for r being Real ex g being Real st ( r < g & g in dom f ) ) & ex g being Real st for seq being Real_Sequence st seq is divergent_to+infty & rng seq c= dom f holds ( f /* seq is convergent & lim (f /* seq) = g ) ); attrf is divergent_in+infty_to+infty means :Def7: :: LIMFUNC1:def 7 ( ( for r being Real ex g being Real st ( r < g & g in dom f ) ) & ( for seq being Real_Sequence st seq is divergent_to+infty & rng seq c= dom f holds f /* seq is divergent_to+infty ) ); attrf is divergent_in+infty_to-infty means :Def8: :: LIMFUNC1:def 8 ( ( for r being Real ex g being Real st ( r < g & g in dom f ) ) & ( for seq being Real_Sequence st seq is divergent_to+infty & rng seq c= dom f holds f /* seq is divergent_to-infty ) ); attrf is convergent_in-infty means :Def9: :: LIMFUNC1:def 9 ( ( for r being Real ex g being Real st ( g < r & g in dom f ) ) & ex g being Real st for seq being Real_Sequence st seq is divergent_to-infty & rng seq c= dom f holds ( f /* seq is convergent & lim (f /* seq) = g ) ); attrf is divergent_in-infty_to+infty means :Def10: :: LIMFUNC1:def 10 ( ( for r being Real ex g being Real st ( g < r & g in dom f ) ) & ( for seq being Real_Sequence st seq is divergent_to-infty & rng seq c= dom f holds f /* seq is divergent_to+infty ) ); attrf is divergent_in-infty_to-infty means :Def11: :: LIMFUNC1:def 11 ( ( for r being Real ex g being Real st ( g < r & g in dom f ) ) & ( for seq being Real_Sequence st seq is divergent_to-infty & rng seq c= dom f holds f /* seq is divergent_to-infty ) ); end; :: deftheorem Def6 defines convergent_in+infty LIMFUNC1:def_6_:_ for f being PartFunc of REAL,REAL holds ( f is convergent_in+infty iff ( ( for r being Real ex g being Real st ( r < g & g in dom f ) ) & ex g being Real st for seq being Real_Sequence st seq is divergent_to+infty & rng seq c= dom f holds ( f /* seq is convergent & lim (f /* seq) = g ) ) ); :: deftheorem Def7 defines divergent_in+infty_to+infty LIMFUNC1:def_7_:_ for f being PartFunc of REAL,REAL holds ( f is divergent_in+infty_to+infty iff ( ( for r being Real ex g being Real st ( r < g & g in dom f ) ) & ( for seq being Real_Sequence st seq is divergent_to+infty & rng seq c= dom f holds f /* seq is divergent_to+infty ) ) ); :: deftheorem Def8 defines divergent_in+infty_to-infty LIMFUNC1:def_8_:_ for f being PartFunc of REAL,REAL holds ( f is divergent_in+infty_to-infty iff ( ( for r being Real ex g being Real st ( r < g & g in dom f ) ) & ( for seq being Real_Sequence st seq is divergent_to+infty & rng seq c= dom f holds f /* seq is divergent_to-infty ) ) ); :: deftheorem Def9 defines convergent_in-infty LIMFUNC1:def_9_:_ for f being PartFunc of REAL,REAL holds ( f is convergent_in-infty iff ( ( for r being Real ex g being Real st ( g < r & g in dom f ) ) & ex g being Real st for seq being Real_Sequence st seq is divergent_to-infty & rng seq c= dom f holds ( f /* seq is convergent & lim (f /* seq) = g ) ) ); :: deftheorem Def10 defines divergent_in-infty_to+infty LIMFUNC1:def_10_:_ for f being PartFunc of REAL,REAL holds ( f is divergent_in-infty_to+infty iff ( ( for r being Real ex g being Real st ( g < r & g in dom f ) ) & ( for seq being Real_Sequence st seq is divergent_to-infty & rng seq c= dom f holds f /* seq is divergent_to+infty ) ) ); :: deftheorem Def11 defines divergent_in-infty_to-infty LIMFUNC1:def_11_:_ for f being PartFunc of REAL,REAL holds ( f is divergent_in-infty_to-infty iff ( ( for r being Real ex g being Real st ( g < r & g in dom f ) ) & ( for seq being Real_Sequence st seq is divergent_to-infty & rng seq c= dom f holds f /* seq is divergent_to-infty ) ) ); theorem :: LIMFUNC1:44 for f being PartFunc of REAL,REAL holds ( f is convergent_in+infty iff ( ( for r being Real ex g being Real st ( r < g & g in dom f ) ) & ex g being Real st for g1 being Real st 0 < g1 holds ex r being Real st for r1 being Real st r < r1 & r1 in dom f holds abs ((f . r1) - g) < g1 ) ) proof let f be PartFunc of REAL,REAL; ::_thesis: ( f is convergent_in+infty iff ( ( for r being Real ex g being Real st ( r < g & g in dom f ) ) & ex g being Real st for g1 being Real st 0 < g1 holds ex r being Real st for r1 being Real st r < r1 & r1 in dom f holds abs ((f . r1) - g) < g1 ) ) thus ( f is convergent_in+infty implies ( ( for r being Real ex g being Real st ( r < g & g in dom f ) ) & ex g being Real st for g1 being Real st 0 < g1 holds ex r being Real st for r1 being Real st r < r1 & r1 in dom f holds abs ((f . r1) - g) < g1 ) ) ::_thesis: ( ( for r being Real ex g being Real st ( r < g & g in dom f ) ) & ex g being Real st for g1 being Real st 0 < g1 holds ex r being Real st for r1 being Real st r < r1 & r1 in dom f holds abs ((f . r1) - g) < g1 implies f is convergent_in+infty ) proof assume A1: f is convergent_in+infty ; ::_thesis: ( ( for r being Real ex g being Real st ( r < g & g in dom f ) ) & ex g being Real st for g1 being Real st 0 < g1 holds ex r being Real st for r1 being Real st r < r1 & r1 in dom f holds abs ((f . r1) - g) < g1 ) then consider g2 being Real such that A2: for seq being Real_Sequence st seq is divergent_to+infty & rng seq c= dom f holds ( f /* seq is convergent & lim (f /* seq) = g2 ) by Def6; assume ( ex r being Real st for g being Real holds ( not r < g or not g in dom f ) or for g being Real ex g1 being Real st ( 0 < g1 & ( for r being Real ex r1 being Real st ( r < r1 & r1 in dom f & abs ((f . r1) - g) >= g1 ) ) ) ) ; ::_thesis: contradiction then consider g being Real such that A3: 0 < g and A4: for r being Real ex r1 being Real st ( r < r1 & r1 in dom f & abs ((f . r1) - g2) >= g ) by A1, Def6; defpred S1[ Element of NAT , real number ] means ( $1 < $2 & $2 in dom f & abs ((f . $2) - g2) >= g ); A5: for n being Element of NAT ex r being Real st S1[n,r] by A4; consider s being Real_Sequence such that A6: for n being Element of NAT holds S1[n,s . n] from FUNCT_2:sch_3(A5); now__::_thesis:_for_x_being_set_st_x_in_rng_s_holds_ x_in_dom_f let x be set ; ::_thesis: ( x in rng s implies x in dom f ) assume x in rng s ; ::_thesis: x in dom f then ex n being Element of NAT st s . n = x by FUNCT_2:113; hence x in dom f by A6; ::_thesis: verum end; then A7: rng s c= dom f by TARSKI:def_3; now__::_thesis:_for_n_being_Element_of_NAT_holds_(incl_NAT)_._n_<=_s_._n let n be Element of NAT ; ::_thesis: (incl NAT) . n <= s . n n < s . n by A6; hence (incl NAT) . n <= s . n by FUNCT_1:18; ::_thesis: verum end; then s is divergent_to+infty by Lm4, Th20, Th42; then ( f /* s is convergent & lim (f /* s) = g2 ) by A2, A7; then consider n being Element of NAT such that A8: for m being Element of NAT st n <= m holds abs (((f /* s) . m) - g2) < g by A3, SEQ_2:def_7; abs (((f /* s) . n) - g2) < g by A8; then abs ((f . (s . n)) - g2) < g by A7, FUNCT_2:108; hence contradiction by A6; ::_thesis: verum end; assume A9: for r being Real ex g being Real st ( r < g & g in dom f ) ; ::_thesis: ( for g being Real ex g1 being Real st ( 0 < g1 & ( for r being Real ex r1 being Real st ( r < r1 & r1 in dom f & not abs ((f . r1) - g) < g1 ) ) ) or f is convergent_in+infty ) given g being Real such that A10: for g1 being Real st 0 < g1 holds ex r being Real st for r1 being Real st r < r1 & r1 in dom f holds abs ((f . r1) - g) < g1 ; ::_thesis: f is convergent_in+infty now__::_thesis:_for_s_being_Real_Sequence_st_s_is_divergent_to+infty_&_rng_s_c=_dom_f_holds_ (_f_/*_s_is_convergent_&_lim_(f_/*_s)_=_g_) let s be Real_Sequence; ::_thesis: ( s is divergent_to+infty & rng s c= dom f implies ( f /* s is convergent & lim (f /* s) = g ) ) assume that A11: s is divergent_to+infty and A12: rng s c= dom f ; ::_thesis: ( f /* s is convergent & lim (f /* s) = g ) A13: now__::_thesis:_for_g1_being_real_number_st_0_<_g1_holds_ ex_n_being_Element_of_NAT_st_ for_m_being_Element_of_NAT_st_n_<=_m_holds_ abs_(((f_/*_s)_._m)_-_g)_<_g1 let g1 be real number ; ::_thesis: ( 0 < g1 implies ex n being Element of NAT st for m being Element of NAT st n <= m holds abs (((f /* s) . m) - g) < g1 ) assume A14: 0 < g1 ; ::_thesis: ex n being Element of NAT st for m being Element of NAT st n <= m holds abs (((f /* s) . m) - g) < g1 g1 is Real by XREAL_0:def_1; then consider r being Real such that A15: for r1 being Real st r < r1 & r1 in dom f holds abs ((f . r1) - g) < g1 by A10, A14; consider n being Element of NAT such that A16: for m being Element of NAT st n <= m holds r < s . m by A11, Def4; take n = n; ::_thesis: for m being Element of NAT st n <= m holds abs (((f /* s) . m) - g) < g1 let m be Element of NAT ; ::_thesis: ( n <= m implies abs (((f /* s) . m) - g) < g1 ) A17: s . m in rng s by VALUED_0:28; assume n <= m ; ::_thesis: abs (((f /* s) . m) - g) < g1 then abs ((f . (s . m)) - g) < g1 by A12, A15, A16, A17; hence abs (((f /* s) . m) - g) < g1 by A12, FUNCT_2:108; ::_thesis: verum end; hence f /* s is convergent by SEQ_2:def_6; ::_thesis: lim (f /* s) = g hence lim (f /* s) = g by A13, SEQ_2:def_7; ::_thesis: verum end; hence f is convergent_in+infty by A9, Def6; ::_thesis: verum end; theorem :: LIMFUNC1:45 for f being PartFunc of REAL,REAL holds ( f is convergent_in-infty iff ( ( for r being Real ex g being Real st ( g < r & g in dom f ) ) & ex g being Real st for g1 being Real st 0 < g1 holds ex r being Real st for r1 being Real st r1 < r & r1 in dom f holds abs ((f . r1) - g) < g1 ) ) proof let f be PartFunc of REAL,REAL; ::_thesis: ( f is convergent_in-infty iff ( ( for r being Real ex g being Real st ( g < r & g in dom f ) ) & ex g being Real st for g1 being Real st 0 < g1 holds ex r being Real st for r1 being Real st r1 < r & r1 in dom f holds abs ((f . r1) - g) < g1 ) ) thus ( f is convergent_in-infty implies ( ( for r being Real ex g being Real st ( g < r & g in dom f ) ) & ex g being Real st for g1 being Real st 0 < g1 holds ex r being Real st for r1 being Real st r1 < r & r1 in dom f holds abs ((f . r1) - g) < g1 ) ) ::_thesis: ( ( for r being Real ex g being Real st ( g < r & g in dom f ) ) & ex g being Real st for g1 being Real st 0 < g1 holds ex r being Real st for r1 being Real st r1 < r & r1 in dom f holds abs ((f . r1) - g) < g1 implies f is convergent_in-infty ) proof assume A1: f is convergent_in-infty ; ::_thesis: ( ( for r being Real ex g being Real st ( g < r & g in dom f ) ) & ex g being Real st for g1 being Real st 0 < g1 holds ex r being Real st for r1 being Real st r1 < r & r1 in dom f holds abs ((f . r1) - g) < g1 ) then consider g2 being Real such that A2: for seq being Real_Sequence st seq is divergent_to-infty & rng seq c= dom f holds ( f /* seq is convergent & lim (f /* seq) = g2 ) by Def9; assume ( ex r being Real st for g being Real holds ( not g < r or not g in dom f ) or for g being Real ex g1 being Real st ( 0 < g1 & ( for r being Real ex r1 being Real st ( r1 < r & r1 in dom f & abs ((f . r1) - g) >= g1 ) ) ) ) ; ::_thesis: contradiction then consider g being Real such that A3: 0 < g and A4: for r being Real ex r1 being Real st ( r1 < r & r1 in dom f & abs ((f . r1) - g2) >= g ) by A1, Def9; defpred S1[ Element of NAT , real number ] means ( $2 < - $1 & $2 in dom f & abs ((f . $2) - g2) >= g ); A5: for n being Element of NAT ex r being Real st S1[n,r] by A4; consider s being Real_Sequence such that A6: for n being Element of NAT holds S1[n,s . n] from FUNCT_2:sch_3(A5); now__::_thesis:_for_x_being_set_st_x_in_rng_s_holds_ x_in_dom_f let x be set ; ::_thesis: ( x in rng s implies x in dom f ) assume x in rng s ; ::_thesis: x in dom f then ex n being Element of NAT st s . n = x by FUNCT_2:113; hence x in dom f by A6; ::_thesis: verum end; then A7: rng s c= dom f by TARSKI:def_3; deffunc H1( Element of NAT ) -> Element of REAL = - $1; consider s1 being Real_Sequence such that A8: for n being Element of NAT holds s1 . n = H1(n) from SEQ_1:sch_1(); now__::_thesis:_for_n_being_Element_of_NAT_holds_s_._n_<=_s1_._n let n be Element of NAT ; ::_thesis: s . n <= s1 . n s . n < - n by A6; hence s . n <= s1 . n by A8; ::_thesis: verum end; then s is divergent_to-infty by A8, Th21, Th43; then ( f /* s is convergent & lim (f /* s) = g2 ) by A2, A7; then consider n being Element of NAT such that A9: for m being Element of NAT st n <= m holds abs (((f /* s) . m) - g2) < g by A3, SEQ_2:def_7; abs (((f /* s) . n) - g2) < g by A9; then abs ((f . (s . n)) - g2) < g by A7, FUNCT_2:108; hence contradiction by A6; ::_thesis: verum end; assume A10: for r being Real ex g being Real st ( g < r & g in dom f ) ; ::_thesis: ( for g being Real ex g1 being Real st ( 0 < g1 & ( for r being Real ex r1 being Real st ( r1 < r & r1 in dom f & not abs ((f . r1) - g) < g1 ) ) ) or f is convergent_in-infty ) given g being Real such that A11: for g1 being Real st 0 < g1 holds ex r being Real st for r1 being Real st r1 < r & r1 in dom f holds abs ((f . r1) - g) < g1 ; ::_thesis: f is convergent_in-infty now__::_thesis:_for_s_being_Real_Sequence_st_s_is_divergent_to-infty_&_rng_s_c=_dom_f_holds_ (_f_/*_s_is_convergent_&_lim_(f_/*_s)_=_g_) let s be Real_Sequence; ::_thesis: ( s is divergent_to-infty & rng s c= dom f implies ( f /* s is convergent & lim (f /* s) = g ) ) assume that A12: s is divergent_to-infty and A13: rng s c= dom f ; ::_thesis: ( f /* s is convergent & lim (f /* s) = g ) A14: now__::_thesis:_for_g1_being_real_number_st_0_<_g1_holds_ ex_n_being_Element_of_NAT_st_ for_m_being_Element_of_NAT_st_n_<=_m_holds_ abs_(((f_/*_s)_._m)_-_g)_<_g1 let g1 be real number ; ::_thesis: ( 0 < g1 implies ex n being Element of NAT st for m being Element of NAT st n <= m holds abs (((f /* s) . m) - g) < g1 ) assume A15: 0 < g1 ; ::_thesis: ex n being Element of NAT st for m being Element of NAT st n <= m holds abs (((f /* s) . m) - g) < g1 g1 is Real by XREAL_0:def_1; then consider r being Real such that A16: for r1 being Real st r1 < r & r1 in dom f holds abs ((f . r1) - g) < g1 by A11, A15; consider n being Element of NAT such that A17: for m being Element of NAT st n <= m holds s . m < r by A12, Def5; take n = n; ::_thesis: for m being Element of NAT st n <= m holds abs (((f /* s) . m) - g) < g1 let m be Element of NAT ; ::_thesis: ( n <= m implies abs (((f /* s) . m) - g) < g1 ) A18: s . m in rng s by VALUED_0:28; assume n <= m ; ::_thesis: abs (((f /* s) . m) - g) < g1 then abs ((f . (s . m)) - g) < g1 by A13, A16, A17, A18; hence abs (((f /* s) . m) - g) < g1 by A13, FUNCT_2:108; ::_thesis: verum end; hence f /* s is convergent by SEQ_2:def_6; ::_thesis: lim (f /* s) = g hence lim (f /* s) = g by A14, SEQ_2:def_7; ::_thesis: verum end; hence f is convergent_in-infty by A10, Def9; ::_thesis: verum end; theorem :: LIMFUNC1:46 for f being PartFunc of REAL,REAL holds ( f is divergent_in+infty_to+infty iff ( ( for r being Real ex g being Real st ( r < g & g in dom f ) ) & ( for g being Real ex r being Real st for r1 being Real st r < r1 & r1 in dom f holds g < f . r1 ) ) ) proof let f be PartFunc of REAL,REAL; ::_thesis: ( f is divergent_in+infty_to+infty iff ( ( for r being Real ex g being Real st ( r < g & g in dom f ) ) & ( for g being Real ex r being Real st for r1 being Real st r < r1 & r1 in dom f holds g < f . r1 ) ) ) thus ( f is divergent_in+infty_to+infty implies ( ( for r being Real ex g being Real st ( r < g & g in dom f ) ) & ( for g being Real ex r being Real st for r1 being Real st r < r1 & r1 in dom f holds g < f . r1 ) ) ) ::_thesis: ( ( for r being Real ex g being Real st ( r < g & g in dom f ) ) & ( for g being Real ex r being Real st for r1 being Real st r < r1 & r1 in dom f holds g < f . r1 ) implies f is divergent_in+infty_to+infty ) proof assume A1: f is divergent_in+infty_to+infty ; ::_thesis: ( ( for r being Real ex g being Real st ( r < g & g in dom f ) ) & ( for g being Real ex r being Real st for r1 being Real st r < r1 & r1 in dom f holds g < f . r1 ) ) assume ( ex r being Real st for g being Real holds ( not r < g or not g in dom f ) or ex g being Real st for r being Real ex r1 being Real st ( r < r1 & r1 in dom f & g >= f . r1 ) ) ; ::_thesis: contradiction then consider g being Real such that A2: for r being Real ex r1 being Real st ( r < r1 & r1 in dom f & g >= f . r1 ) by A1, Def7; defpred S1[ Element of NAT , real number ] means ( $1 < $2 & $2 in dom f & g >= f . $2 ); A3: for n being Element of NAT ex r being Real st S1[n,r] by A2; consider s being Real_Sequence such that A4: for n being Element of NAT holds S1[n,s . n] from FUNCT_2:sch_3(A3); now__::_thesis:_for_x_being_set_st_x_in_rng_s_holds_ x_in_dom_f let x be set ; ::_thesis: ( x in rng s implies x in dom f ) assume x in rng s ; ::_thesis: x in dom f then ex n being Element of NAT st s . n = x by FUNCT_2:113; hence x in dom f by A4; ::_thesis: verum end; then A5: rng s c= dom f by TARSKI:def_3; now__::_thesis:_for_n_being_Element_of_NAT_holds_(incl_NAT)_._n_<=_s_._n let n be Element of NAT ; ::_thesis: (incl NAT) . n <= s . n n < s . n by A4; hence (incl NAT) . n <= s . n by FUNCT_1:18; ::_thesis: verum end; then s is divergent_to+infty by Lm4, Th20, Th42; then f /* s is divergent_to+infty by A1, A5, Def7; then consider n being Element of NAT such that A6: for m being Element of NAT st n <= m holds g < (f /* s) . m by Def4; g < (f /* s) . n by A6; then g < f . (s . n) by A5, FUNCT_2:108; hence contradiction by A4; ::_thesis: verum end; assume that A7: for r being Real ex g being Real st ( r < g & g in dom f ) and A8: for g being Real ex r being Real st for r1 being Real st r < r1 & r1 in dom f holds g < f . r1 ; ::_thesis: f is divergent_in+infty_to+infty now__::_thesis:_for_s_being_Real_Sequence_st_s_is_divergent_to+infty_&_rng_s_c=_dom_f_holds_ f_/*_s_is_divergent_to+infty let s be Real_Sequence; ::_thesis: ( s is divergent_to+infty & rng s c= dom f implies f /* s is divergent_to+infty ) assume that A9: s is divergent_to+infty and A10: rng s c= dom f ; ::_thesis: f /* s is divergent_to+infty now__::_thesis:_for_g_being_Real_ex_n_being_Element_of_NAT_st_ for_m_being_Element_of_NAT_st_n_<=_m_holds_ g_<_(f_/*_s)_._m let g be Real; ::_thesis: ex n being Element of NAT st for m being Element of NAT st n <= m holds g < (f /* s) . m consider r being Real such that A11: for r1 being Real st r < r1 & r1 in dom f holds g < f . r1 by A8; consider n being Element of NAT such that A12: for m being Element of NAT st n <= m holds r < s . m by A9, Def4; take n = n; ::_thesis: for m being Element of NAT st n <= m holds g < (f /* s) . m let m be Element of NAT ; ::_thesis: ( n <= m implies g < (f /* s) . m ) A13: s . m in rng s by VALUED_0:28; assume n <= m ; ::_thesis: g < (f /* s) . m then g < f . (s . m) by A10, A11, A12, A13; hence g < (f /* s) . m by A10, FUNCT_2:108; ::_thesis: verum end; hence f /* s is divergent_to+infty by Def4; ::_thesis: verum end; hence f is divergent_in+infty_to+infty by A7, Def7; ::_thesis: verum end; theorem :: LIMFUNC1:47 for f being PartFunc of REAL,REAL holds ( f is divergent_in+infty_to-infty iff ( ( for r being Real ex g being Real st ( r < g & g in dom f ) ) & ( for g being Real ex r being Real st for r1 being Real st r < r1 & r1 in dom f holds f . r1 < g ) ) ) proof let f be PartFunc of REAL,REAL; ::_thesis: ( f is divergent_in+infty_to-infty iff ( ( for r being Real ex g being Real st ( r < g & g in dom f ) ) & ( for g being Real ex r being Real st for r1 being Real st r < r1 & r1 in dom f holds f . r1 < g ) ) ) thus ( f is divergent_in+infty_to-infty implies ( ( for r being Real ex g being Real st ( r < g & g in dom f ) ) & ( for g being Real ex r being Real st for r1 being Real st r < r1 & r1 in dom f holds f . r1 < g ) ) ) ::_thesis: ( ( for r being Real ex g being Real st ( r < g & g in dom f ) ) & ( for g being Real ex r being Real st for r1 being Real st r < r1 & r1 in dom f holds f . r1 < g ) implies f is divergent_in+infty_to-infty ) proof assume A1: f is divergent_in+infty_to-infty ; ::_thesis: ( ( for r being Real ex g being Real st ( r < g & g in dom f ) ) & ( for g being Real ex r being Real st for r1 being Real st r < r1 & r1 in dom f holds f . r1 < g ) ) assume ( ex r being Real st for g being Real holds ( not r < g or not g in dom f ) or ex g being Real st for r being Real ex r1 being Real st ( r < r1 & r1 in dom f & f . r1 >= g ) ) ; ::_thesis: contradiction then consider g being Real such that A2: for r being Real ex r1 being Real st ( r < r1 & r1 in dom f & f . r1 >= g ) by A1, Def8; defpred S1[ Element of NAT , real number ] means ( $1 < $2 & $2 in dom f & g <= f . $2 ); A3: for n being Element of NAT ex r being Real st S1[n,r] by A2; consider s being Real_Sequence such that A4: for n being Element of NAT holds S1[n,s . n] from FUNCT_2:sch_3(A3); now__::_thesis:_for_x_being_set_st_x_in_rng_s_holds_ x_in_dom_f let x be set ; ::_thesis: ( x in rng s implies x in dom f ) assume x in rng s ; ::_thesis: x in dom f then ex n being Element of NAT st s . n = x by FUNCT_2:113; hence x in dom f by A4; ::_thesis: verum end; then A5: rng s c= dom f by TARSKI:def_3; now__::_thesis:_for_n_being_Element_of_NAT_holds_(incl_NAT)_._n_<=_s_._n let n be Element of NAT ; ::_thesis: (incl NAT) . n <= s . n n < s . n by A4; hence (incl NAT) . n <= s . n by FUNCT_1:18; ::_thesis: verum end; then s is divergent_to+infty by Lm4, Th20, Th42; then f /* s is divergent_to-infty by A1, A5, Def8; then consider n being Element of NAT such that A6: for m being Element of NAT st n <= m holds (f /* s) . m < g by Def5; (f /* s) . n < g by A6; then f . (s . n) < g by A5, FUNCT_2:108; hence contradiction by A4; ::_thesis: verum end; assume that A7: for r being Real ex g being Real st ( r < g & g in dom f ) and A8: for g being Real ex r being Real st for r1 being Real st r < r1 & r1 in dom f holds f . r1 < g ; ::_thesis: f is divergent_in+infty_to-infty now__::_thesis:_for_s_being_Real_Sequence_st_s_is_divergent_to+infty_&_rng_s_c=_dom_f_holds_ f_/*_s_is_divergent_to-infty let s be Real_Sequence; ::_thesis: ( s is divergent_to+infty & rng s c= dom f implies f /* s is divergent_to-infty ) assume that A9: s is divergent_to+infty and A10: rng s c= dom f ; ::_thesis: f /* s is divergent_to-infty now__::_thesis:_for_g_being_Real_ex_n_being_Element_of_NAT_st_ for_m_being_Element_of_NAT_st_n_<=_m_holds_ (f_/*_s)_._m_<_g let g be Real; ::_thesis: ex n being Element of NAT st for m being Element of NAT st n <= m holds (f /* s) . m < g consider r being Real such that A11: for r1 being Real st r < r1 & r1 in dom f holds f . r1 < g by A8; consider n being Element of NAT such that A12: for m being Element of NAT st n <= m holds r < s . m by A9, Def4; take n = n; ::_thesis: for m being Element of NAT st n <= m holds (f /* s) . m < g let m be Element of NAT ; ::_thesis: ( n <= m implies (f /* s) . m < g ) A13: s . m in rng s by VALUED_0:28; assume n <= m ; ::_thesis: (f /* s) . m < g then f . (s . m) < g by A10, A11, A12, A13; hence (f /* s) . m < g by A10, FUNCT_2:108; ::_thesis: verum end; hence f /* s is divergent_to-infty by Def5; ::_thesis: verum end; hence f is divergent_in+infty_to-infty by A7, Def8; ::_thesis: verum end; theorem :: LIMFUNC1:48 for f being PartFunc of REAL,REAL holds ( f is divergent_in-infty_to+infty iff ( ( for r being Real ex g being Real st ( g < r & g in dom f ) ) & ( for g being Real ex r being Real st for r1 being Real st r1 < r & r1 in dom f holds g < f . r1 ) ) ) proof let f be PartFunc of REAL,REAL; ::_thesis: ( f is divergent_in-infty_to+infty iff ( ( for r being Real ex g being Real st ( g < r & g in dom f ) ) & ( for g being Real ex r being Real st for r1 being Real st r1 < r & r1 in dom f holds g < f . r1 ) ) ) thus ( f is divergent_in-infty_to+infty implies ( ( for r being Real ex g being Real st ( g < r & g in dom f ) ) & ( for g being Real ex r being Real st for r1 being Real st r1 < r & r1 in dom f holds g < f . r1 ) ) ) ::_thesis: ( ( for r being Real ex g being Real st ( g < r & g in dom f ) ) & ( for g being Real ex r being Real st for r1 being Real st r1 < r & r1 in dom f holds g < f . r1 ) implies f is divergent_in-infty_to+infty ) proof deffunc H1( Element of NAT ) -> Element of REAL = - $1; assume A1: f is divergent_in-infty_to+infty ; ::_thesis: ( ( for r being Real ex g being Real st ( g < r & g in dom f ) ) & ( for g being Real ex r being Real st for r1 being Real st r1 < r & r1 in dom f holds g < f . r1 ) ) assume ( ex r being Real st for g being Real holds ( not g < r or not g in dom f ) or ex g being Real st for r being Real ex r1 being Real st ( r1 < r & r1 in dom f & g >= f . r1 ) ) ; ::_thesis: contradiction then consider g being Real such that A2: for r being Real ex r1 being Real st ( r1 < r & r1 in dom f & g >= f . r1 ) by A1, Def10; defpred S1[ Element of NAT , real number ] means ( $2 < - $1 & $2 in dom f & g >= f . $2 ); A3: for n being Element of NAT ex r being Real st S1[n,r] by A2; consider s being Real_Sequence such that A4: for n being Element of NAT holds S1[n,s . n] from FUNCT_2:sch_3(A3); now__::_thesis:_for_x_being_set_st_x_in_rng_s_holds_ x_in_dom_f let x be set ; ::_thesis: ( x in rng s implies x in dom f ) assume x in rng s ; ::_thesis: x in dom f then ex n being Element of NAT st s . n = x by FUNCT_2:113; hence x in dom f by A4; ::_thesis: verum end; then A5: rng s c= dom f by TARSKI:def_3; consider s1 being Real_Sequence such that A6: for n being Element of NAT holds s1 . n = H1(n) from SEQ_1:sch_1(); now__::_thesis:_for_n_being_Element_of_NAT_holds_s_._n_<=_s1_._n let n be Element of NAT ; ::_thesis: s . n <= s1 . n s . n < - n by A4; hence s . n <= s1 . n by A6; ::_thesis: verum end; then s is divergent_to-infty by A6, Th21, Th43; then f /* s is divergent_to+infty by A1, A5, Def10; then consider n being Element of NAT such that A7: for m being Element of NAT st n <= m holds g < (f /* s) . m by Def4; g < (f /* s) . n by A7; then g < f . (s . n) by A5, FUNCT_2:108; hence contradiction by A4; ::_thesis: verum end; assume that A8: for r being Real ex g being Real st ( g < r & g in dom f ) and A9: for g being Real ex r being Real st for r1 being Real st r1 < r & r1 in dom f holds g < f . r1 ; ::_thesis: f is divergent_in-infty_to+infty now__::_thesis:_for_s_being_Real_Sequence_st_s_is_divergent_to-infty_&_rng_s_c=_dom_f_holds_ f_/*_s_is_divergent_to+infty let s be Real_Sequence; ::_thesis: ( s is divergent_to-infty & rng s c= dom f implies f /* s is divergent_to+infty ) assume that A10: s is divergent_to-infty and A11: rng s c= dom f ; ::_thesis: f /* s is divergent_to+infty now__::_thesis:_for_g_being_Real_ex_n_being_Element_of_NAT_st_ for_m_being_Element_of_NAT_st_n_<=_m_holds_ g_<_(f_/*_s)_._m let g be Real; ::_thesis: ex n being Element of NAT st for m being Element of NAT st n <= m holds g < (f /* s) . m consider r being Real such that A12: for r1 being Real st r1 < r & r1 in dom f holds g < f . r1 by A9; consider n being Element of NAT such that A13: for m being Element of NAT st n <= m holds s . m < r by A10, Def5; take n = n; ::_thesis: for m being Element of NAT st n <= m holds g < (f /* s) . m let m be Element of NAT ; ::_thesis: ( n <= m implies g < (f /* s) . m ) A14: s . m in rng s by VALUED_0:28; assume n <= m ; ::_thesis: g < (f /* s) . m then g < f . (s . m) by A11, A12, A13, A14; hence g < (f /* s) . m by A11, FUNCT_2:108; ::_thesis: verum end; hence f /* s is divergent_to+infty by Def4; ::_thesis: verum end; hence f is divergent_in-infty_to+infty by A8, Def10; ::_thesis: verum end; theorem :: LIMFUNC1:49 for f being PartFunc of REAL,REAL holds ( f is divergent_in-infty_to-infty iff ( ( for r being Real ex g being Real st ( g < r & g in dom f ) ) & ( for g being Real ex r being Real st for r1 being Real st r1 < r & r1 in dom f holds f . r1 < g ) ) ) proof let f be PartFunc of REAL,REAL; ::_thesis: ( f is divergent_in-infty_to-infty iff ( ( for r being Real ex g being Real st ( g < r & g in dom f ) ) & ( for g being Real ex r being Real st for r1 being Real st r1 < r & r1 in dom f holds f . r1 < g ) ) ) thus ( f is divergent_in-infty_to-infty implies ( ( for r being Real ex g being Real st ( g < r & g in dom f ) ) & ( for g being Real ex r being Real st for r1 being Real st r1 < r & r1 in dom f holds f . r1 < g ) ) ) ::_thesis: ( ( for r being Real ex g being Real st ( g < r & g in dom f ) ) & ( for g being Real ex r being Real st for r1 being Real st r1 < r & r1 in dom f holds f . r1 < g ) implies f is divergent_in-infty_to-infty ) proof deffunc H1( Element of NAT ) -> Element of REAL = - $1; assume A1: f is divergent_in-infty_to-infty ; ::_thesis: ( ( for r being Real ex g being Real st ( g < r & g in dom f ) ) & ( for g being Real ex r being Real st for r1 being Real st r1 < r & r1 in dom f holds f . r1 < g ) ) assume ( ex r being Real st for g being Real holds ( not g < r or not g in dom f ) or ex g being Real st for r being Real ex r1 being Real st ( r1 < r & r1 in dom f & f . r1 >= g ) ) ; ::_thesis: contradiction then consider g being Real such that A2: for r being Real ex r1 being Real st ( r1 < r & r1 in dom f & f . r1 >= g ) by A1, Def11; defpred S1[ Element of NAT , real number ] means ( $2 < - $1 & $2 in dom f & g <= f . $2 ); A3: for n being Element of NAT ex r being Real st S1[n,r] by A2; consider s being Real_Sequence such that A4: for n being Element of NAT holds S1[n,s . n] from FUNCT_2:sch_3(A3); now__::_thesis:_for_x_being_set_st_x_in_rng_s_holds_ x_in_dom_f let x be set ; ::_thesis: ( x in rng s implies x in dom f ) assume x in rng s ; ::_thesis: x in dom f then ex n being Element of NAT st s . n = x by FUNCT_2:113; hence x in dom f by A4; ::_thesis: verum end; then A5: rng s c= dom f by TARSKI:def_3; consider s1 being Real_Sequence such that A6: for n being Element of NAT holds s1 . n = H1(n) from SEQ_1:sch_1(); now__::_thesis:_for_n_being_Element_of_NAT_holds_s_._n_<=_s1_._n let n be Element of NAT ; ::_thesis: s . n <= s1 . n s . n < - n by A4; hence s . n <= s1 . n by A6; ::_thesis: verum end; then s is divergent_to-infty by A6, Th21, Th43; then f /* s is divergent_to-infty by A1, A5, Def11; then consider n being Element of NAT such that A7: for m being Element of NAT st n <= m holds (f /* s) . m < g by Def5; (f /* s) . n < g by A7; then f . (s . n) < g by A5, FUNCT_2:108; hence contradiction by A4; ::_thesis: verum end; assume that A8: for r being Real ex g being Real st ( g < r & g in dom f ) and A9: for g being Real ex r being Real st for r1 being Real st r1 < r & r1 in dom f holds f . r1 < g ; ::_thesis: f is divergent_in-infty_to-infty now__::_thesis:_for_s_being_Real_Sequence_st_s_is_divergent_to-infty_&_rng_s_c=_dom_f_holds_ f_/*_s_is_divergent_to-infty let s be Real_Sequence; ::_thesis: ( s is divergent_to-infty & rng s c= dom f implies f /* s is divergent_to-infty ) assume that A10: s is divergent_to-infty and A11: rng s c= dom f ; ::_thesis: f /* s is divergent_to-infty now__::_thesis:_for_g_being_Real_ex_n_being_Element_of_NAT_st_ for_m_being_Element_of_NAT_st_n_<=_m_holds_ (f_/*_s)_._m_<_g let g be Real; ::_thesis: ex n being Element of NAT st for m being Element of NAT st n <= m holds (f /* s) . m < g consider r being Real such that A12: for r1 being Real st r1 < r & r1 in dom f holds f . r1 < g by A9; consider n being Element of NAT such that A13: for m being Element of NAT st n <= m holds s . m < r by A10, Def5; take n = n; ::_thesis: for m being Element of NAT st n <= m holds (f /* s) . m < g let m be Element of NAT ; ::_thesis: ( n <= m implies (f /* s) . m < g ) A14: s . m in rng s by VALUED_0:28; assume n <= m ; ::_thesis: (f /* s) . m < g then f . (s . m) < g by A11, A12, A13, A14; hence (f /* s) . m < g by A11, FUNCT_2:108; ::_thesis: verum end; hence f /* s is divergent_to-infty by Def5; ::_thesis: verum end; hence f is divergent_in-infty_to-infty by A8, Def11; ::_thesis: verum end; theorem :: LIMFUNC1:50 for f1, f2 being PartFunc of REAL,REAL st f1 is divergent_in+infty_to+infty & f2 is divergent_in+infty_to+infty & ( for r being Real ex g being Real st ( r < g & g in (dom f1) /\ (dom f2) ) ) holds ( f1 + f2 is divergent_in+infty_to+infty & f1 (#) f2 is divergent_in+infty_to+infty ) proof let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is divergent_in+infty_to+infty & f2 is divergent_in+infty_to+infty & ( for r being Real ex g being Real st ( r < g & g in (dom f1) /\ (dom f2) ) ) implies ( f1 + f2 is divergent_in+infty_to+infty & f1 (#) f2 is divergent_in+infty_to+infty ) ) assume that A1: f1 is divergent_in+infty_to+infty and A2: f2 is divergent_in+infty_to+infty and A3: for r being Real ex g being Real st ( r < g & g in (dom f1) /\ (dom f2) ) ; ::_thesis: ( f1 + f2 is divergent_in+infty_to+infty & f1 (#) f2 is divergent_in+infty_to+infty ) A4: now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_divergent_to+infty_&_rng_seq_c=_dom_(f1_+_f2)_holds_ (f1_+_f2)_/*_seq_is_divergent_to+infty let seq be Real_Sequence; ::_thesis: ( seq is divergent_to+infty & rng seq c= dom (f1 + f2) implies (f1 + f2) /* seq is divergent_to+infty ) assume that A5: seq is divergent_to+infty and A6: rng seq c= dom (f1 + f2) ; ::_thesis: (f1 + f2) /* seq is divergent_to+infty A7: dom (f1 + f2) = (dom f1) /\ (dom f2) by A6, Lm2; rng seq c= dom f2 by A6, Lm2; then A8: f2 /* seq is divergent_to+infty by A2, A5, Def7; rng seq c= dom f1 by A6, Lm2; then f1 /* seq is divergent_to+infty by A1, A5, Def7; then (f1 /* seq) + (f2 /* seq) is divergent_to+infty by A8, Th8; hence (f1 + f2) /* seq is divergent_to+infty by A6, A7, RFUNCT_2:8; ::_thesis: verum end; A9: now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_divergent_to+infty_&_rng_seq_c=_dom_(f1_(#)_f2)_holds_ (f1_(#)_f2)_/*_seq_is_divergent_to+infty let seq be Real_Sequence; ::_thesis: ( seq is divergent_to+infty & rng seq c= dom (f1 (#) f2) implies (f1 (#) f2) /* seq is divergent_to+infty ) assume that A10: seq is divergent_to+infty and A11: rng seq c= dom (f1 (#) f2) ; ::_thesis: (f1 (#) f2) /* seq is divergent_to+infty A12: dom (f1 (#) f2) = (dom f1) /\ (dom f2) by A11, Lm3; rng seq c= dom f2 by A11, Lm3; then A13: f2 /* seq is divergent_to+infty by A2, A10, Def7; rng seq c= dom f1 by A11, Lm3; then f1 /* seq is divergent_to+infty by A1, A10, Def7; then (f1 /* seq) (#) (f2 /* seq) is divergent_to+infty by A13, Th10; hence (f1 (#) f2) /* seq is divergent_to+infty by A11, A12, RFUNCT_2:8; ::_thesis: verum end; now__::_thesis:_for_r_being_Real_ex_g_being_Real_st_ (_r_<_g_&_g_in_dom_(f1_+_f2)_) let r be Real; ::_thesis: ex g being Real st ( r < g & g in dom (f1 + f2) ) consider g being Real such that A14: ( r < g & g in (dom f1) /\ (dom f2) ) by A3; take g = g; ::_thesis: ( r < g & g in dom (f1 + f2) ) thus ( r < g & g in dom (f1 + f2) ) by A14, VALUED_1:def_1; ::_thesis: verum end; hence f1 + f2 is divergent_in+infty_to+infty by A4, Def7; ::_thesis: f1 (#) f2 is divergent_in+infty_to+infty now__::_thesis:_for_r_being_Real_ex_g_being_Real_st_ (_r_<_g_&_g_in_dom_(f1_(#)_f2)_) let r be Real; ::_thesis: ex g being Real st ( r < g & g in dom (f1 (#) f2) ) consider g being Real such that A15: ( r < g & g in (dom f1) /\ (dom f2) ) by A3; take g = g; ::_thesis: ( r < g & g in dom (f1 (#) f2) ) thus ( r < g & g in dom (f1 (#) f2) ) by A15, VALUED_1:def_4; ::_thesis: verum end; hence f1 (#) f2 is divergent_in+infty_to+infty by A9, Def7; ::_thesis: verum end; theorem :: LIMFUNC1:51 for f1, f2 being PartFunc of REAL,REAL st f1 is divergent_in+infty_to-infty & f2 is divergent_in+infty_to-infty & ( for r being Real ex g being Real st ( r < g & g in (dom f1) /\ (dom f2) ) ) holds ( f1 + f2 is divergent_in+infty_to-infty & f1 (#) f2 is divergent_in+infty_to+infty ) proof let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is divergent_in+infty_to-infty & f2 is divergent_in+infty_to-infty & ( for r being Real ex g being Real st ( r < g & g in (dom f1) /\ (dom f2) ) ) implies ( f1 + f2 is divergent_in+infty_to-infty & f1 (#) f2 is divergent_in+infty_to+infty ) ) assume that A1: f1 is divergent_in+infty_to-infty and A2: f2 is divergent_in+infty_to-infty and A3: for r being Real ex g being Real st ( r < g & g in (dom f1) /\ (dom f2) ) ; ::_thesis: ( f1 + f2 is divergent_in+infty_to-infty & f1 (#) f2 is divergent_in+infty_to+infty ) A4: now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_divergent_to+infty_&_rng_seq_c=_dom_(f1_+_f2)_holds_ (f1_+_f2)_/*_seq_is_divergent_to-infty let seq be Real_Sequence; ::_thesis: ( seq is divergent_to+infty & rng seq c= dom (f1 + f2) implies (f1 + f2) /* seq is divergent_to-infty ) assume that A5: seq is divergent_to+infty and A6: rng seq c= dom (f1 + f2) ; ::_thesis: (f1 + f2) /* seq is divergent_to-infty A7: dom (f1 + f2) = (dom f1) /\ (dom f2) by A6, Lm2; rng seq c= dom f2 by A6, Lm2; then A8: f2 /* seq is divergent_to-infty by A2, A5, Def8; rng seq c= dom f1 by A6, Lm2; then f1 /* seq is divergent_to-infty by A1, A5, Def8; then (f1 /* seq) + (f2 /* seq) is divergent_to-infty by A8, Th11; hence (f1 + f2) /* seq is divergent_to-infty by A6, A7, RFUNCT_2:8; ::_thesis: verum end; A9: now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_divergent_to+infty_&_rng_seq_c=_dom_(f1_(#)_f2)_holds_ (f1_(#)_f2)_/*_seq_is_divergent_to+infty let seq be Real_Sequence; ::_thesis: ( seq is divergent_to+infty & rng seq c= dom (f1 (#) f2) implies (f1 (#) f2) /* seq is divergent_to+infty ) assume that A10: seq is divergent_to+infty and A11: rng seq c= dom (f1 (#) f2) ; ::_thesis: (f1 (#) f2) /* seq is divergent_to+infty A12: dom (f1 (#) f2) = (dom f1) /\ (dom f2) by A11, Lm3; rng seq c= dom f2 by A11, Lm3; then A13: f2 /* seq is divergent_to-infty by A2, A10, Def8; rng seq c= dom f1 by A11, Lm3; then f1 /* seq is divergent_to-infty by A1, A10, Def8; then (f1 /* seq) (#) (f2 /* seq) is divergent_to+infty by A13, Th24; hence (f1 (#) f2) /* seq is divergent_to+infty by A11, A12, RFUNCT_2:8; ::_thesis: verum end; now__::_thesis:_for_r_being_Real_ex_g_being_Real_st_ (_r_<_g_&_g_in_dom_(f1_+_f2)_) let r be Real; ::_thesis: ex g being Real st ( r < g & g in dom (f1 + f2) ) consider g being Real such that A14: ( r < g & g in (dom f1) /\ (dom f2) ) by A3; take g = g; ::_thesis: ( r < g & g in dom (f1 + f2) ) thus ( r < g & g in dom (f1 + f2) ) by A14, VALUED_1:def_1; ::_thesis: verum end; hence f1 + f2 is divergent_in+infty_to-infty by A4, Def8; ::_thesis: f1 (#) f2 is divergent_in+infty_to+infty now__::_thesis:_for_r_being_Real_ex_g_being_Real_st_ (_r_<_g_&_g_in_dom_(f1_(#)_f2)_) let r be Real; ::_thesis: ex g being Real st ( r < g & g in dom (f1 (#) f2) ) consider g being Real such that A15: ( r < g & g in (dom f1) /\ (dom f2) ) by A3; take g = g; ::_thesis: ( r < g & g in dom (f1 (#) f2) ) thus ( r < g & g in dom (f1 (#) f2) ) by A15, VALUED_1:def_4; ::_thesis: verum end; hence f1 (#) f2 is divergent_in+infty_to+infty by A9, Def7; ::_thesis: verum end; theorem :: LIMFUNC1:52 for f1, f2 being PartFunc of REAL,REAL st f1 is divergent_in-infty_to+infty & f2 is divergent_in-infty_to+infty & ( for r being Real ex g being Real st ( g < r & g in (dom f1) /\ (dom f2) ) ) holds ( f1 + f2 is divergent_in-infty_to+infty & f1 (#) f2 is divergent_in-infty_to+infty ) proof let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is divergent_in-infty_to+infty & f2 is divergent_in-infty_to+infty & ( for r being Real ex g being Real st ( g < r & g in (dom f1) /\ (dom f2) ) ) implies ( f1 + f2 is divergent_in-infty_to+infty & f1 (#) f2 is divergent_in-infty_to+infty ) ) assume that A1: f1 is divergent_in-infty_to+infty and A2: f2 is divergent_in-infty_to+infty and A3: for r being Real ex g being Real st ( g < r & g in (dom f1) /\ (dom f2) ) ; ::_thesis: ( f1 + f2 is divergent_in-infty_to+infty & f1 (#) f2 is divergent_in-infty_to+infty ) A4: now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_divergent_to-infty_&_rng_seq_c=_dom_(f1_+_f2)_holds_ (f1_+_f2)_/*_seq_is_divergent_to+infty let seq be Real_Sequence; ::_thesis: ( seq is divergent_to-infty & rng seq c= dom (f1 + f2) implies (f1 + f2) /* seq is divergent_to+infty ) assume that A5: seq is divergent_to-infty and A6: rng seq c= dom (f1 + f2) ; ::_thesis: (f1 + f2) /* seq is divergent_to+infty A7: dom (f1 + f2) = (dom f1) /\ (dom f2) by A6, Lm2; rng seq c= dom f2 by A6, Lm2; then A8: f2 /* seq is divergent_to+infty by A2, A5, Def10; rng seq c= dom f1 by A6, Lm2; then f1 /* seq is divergent_to+infty by A1, A5, Def10; then (f1 /* seq) + (f2 /* seq) is divergent_to+infty by A8, Th8; hence (f1 + f2) /* seq is divergent_to+infty by A6, A7, RFUNCT_2:8; ::_thesis: verum end; A9: now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_divergent_to-infty_&_rng_seq_c=_dom_(f1_(#)_f2)_holds_ (f1_(#)_f2)_/*_seq_is_divergent_to+infty let seq be Real_Sequence; ::_thesis: ( seq is divergent_to-infty & rng seq c= dom (f1 (#) f2) implies (f1 (#) f2) /* seq is divergent_to+infty ) assume that A10: seq is divergent_to-infty and A11: rng seq c= dom (f1 (#) f2) ; ::_thesis: (f1 (#) f2) /* seq is divergent_to+infty A12: dom (f1 (#) f2) = (dom f1) /\ (dom f2) by A11, Lm3; rng seq c= dom f2 by A11, Lm3; then A13: f2 /* seq is divergent_to+infty by A2, A10, Def10; rng seq c= dom f1 by A11, Lm3; then f1 /* seq is divergent_to+infty by A1, A10, Def10; then (f1 /* seq) (#) (f2 /* seq) is divergent_to+infty by A13, Th10; hence (f1 (#) f2) /* seq is divergent_to+infty by A11, A12, RFUNCT_2:8; ::_thesis: verum end; now__::_thesis:_for_r_being_Real_ex_g_being_Real_st_ (_g_<_r_&_g_in_dom_(f1_+_f2)_) let r be Real; ::_thesis: ex g being Real st ( g < r & g in dom (f1 + f2) ) consider g being Real such that A14: ( g < r & g in (dom f1) /\ (dom f2) ) by A3; take g = g; ::_thesis: ( g < r & g in dom (f1 + f2) ) thus ( g < r & g in dom (f1 + f2) ) by A14, VALUED_1:def_1; ::_thesis: verum end; hence f1 + f2 is divergent_in-infty_to+infty by A4, Def10; ::_thesis: f1 (#) f2 is divergent_in-infty_to+infty now__::_thesis:_for_r_being_Real_ex_g_being_Real_st_ (_g_<_r_&_g_in_dom_(f1_(#)_f2)_) let r be Real; ::_thesis: ex g being Real st ( g < r & g in dom (f1 (#) f2) ) consider g being Real such that A15: ( g < r & g in (dom f1) /\ (dom f2) ) by A3; take g = g; ::_thesis: ( g < r & g in dom (f1 (#) f2) ) thus ( g < r & g in dom (f1 (#) f2) ) by A15, VALUED_1:def_4; ::_thesis: verum end; hence f1 (#) f2 is divergent_in-infty_to+infty by A9, Def10; ::_thesis: verum end; theorem :: LIMFUNC1:53 for f1, f2 being PartFunc of REAL,REAL st f1 is divergent_in-infty_to-infty & f2 is divergent_in-infty_to-infty & ( for r being Real ex g being Real st ( g < r & g in (dom f1) /\ (dom f2) ) ) holds ( f1 + f2 is divergent_in-infty_to-infty & f1 (#) f2 is divergent_in-infty_to+infty ) proof let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is divergent_in-infty_to-infty & f2 is divergent_in-infty_to-infty & ( for r being Real ex g being Real st ( g < r & g in (dom f1) /\ (dom f2) ) ) implies ( f1 + f2 is divergent_in-infty_to-infty & f1 (#) f2 is divergent_in-infty_to+infty ) ) assume that A1: f1 is divergent_in-infty_to-infty and A2: f2 is divergent_in-infty_to-infty and A3: for r being Real ex g being Real st ( g < r & g in (dom f1) /\ (dom f2) ) ; ::_thesis: ( f1 + f2 is divergent_in-infty_to-infty & f1 (#) f2 is divergent_in-infty_to+infty ) A4: now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_divergent_to-infty_&_rng_seq_c=_dom_(f1_+_f2)_holds_ (f1_+_f2)_/*_seq_is_divergent_to-infty let seq be Real_Sequence; ::_thesis: ( seq is divergent_to-infty & rng seq c= dom (f1 + f2) implies (f1 + f2) /* seq is divergent_to-infty ) assume that A5: seq is divergent_to-infty and A6: rng seq c= dom (f1 + f2) ; ::_thesis: (f1 + f2) /* seq is divergent_to-infty A7: dom (f1 + f2) = (dom f1) /\ (dom f2) by A6, Lm2; rng seq c= dom f2 by A6, Lm2; then A8: f2 /* seq is divergent_to-infty by A2, A5, Def11; rng seq c= dom f1 by A6, Lm2; then f1 /* seq is divergent_to-infty by A1, A5, Def11; then (f1 /* seq) + (f2 /* seq) is divergent_to-infty by A8, Th11; hence (f1 + f2) /* seq is divergent_to-infty by A6, A7, RFUNCT_2:8; ::_thesis: verum end; A9: now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_divergent_to-infty_&_rng_seq_c=_dom_(f1_(#)_f2)_holds_ (f1_(#)_f2)_/*_seq_is_divergent_to+infty let seq be Real_Sequence; ::_thesis: ( seq is divergent_to-infty & rng seq c= dom (f1 (#) f2) implies (f1 (#) f2) /* seq is divergent_to+infty ) assume that A10: seq is divergent_to-infty and A11: rng seq c= dom (f1 (#) f2) ; ::_thesis: (f1 (#) f2) /* seq is divergent_to+infty A12: dom (f1 (#) f2) = (dom f1) /\ (dom f2) by A11, Lm3; rng seq c= dom f2 by A11, Lm3; then A13: f2 /* seq is divergent_to-infty by A2, A10, Def11; rng seq c= dom f1 by A11, Lm3; then f1 /* seq is divergent_to-infty by A1, A10, Def11; then (f1 /* seq) (#) (f2 /* seq) is divergent_to+infty by A13, Th24; hence (f1 (#) f2) /* seq is divergent_to+infty by A11, A12, RFUNCT_2:8; ::_thesis: verum end; now__::_thesis:_for_r_being_Real_ex_g_being_Real_st_ (_g_<_r_&_g_in_dom_(f1_+_f2)_) let r be Real; ::_thesis: ex g being Real st ( g < r & g in dom (f1 + f2) ) consider g being Real such that A14: ( g < r & g in (dom f1) /\ (dom f2) ) by A3; take g = g; ::_thesis: ( g < r & g in dom (f1 + f2) ) thus ( g < r & g in dom (f1 + f2) ) by A14, VALUED_1:def_1; ::_thesis: verum end; hence f1 + f2 is divergent_in-infty_to-infty by A4, Def11; ::_thesis: f1 (#) f2 is divergent_in-infty_to+infty now__::_thesis:_for_r_being_Real_ex_g_being_Real_st_ (_g_<_r_&_g_in_dom_(f1_(#)_f2)_) let r be Real; ::_thesis: ex g being Real st ( g < r & g in dom (f1 (#) f2) ) consider g being Real such that A15: ( g < r & g in (dom f1) /\ (dom f2) ) by A3; take g = g; ::_thesis: ( g < r & g in dom (f1 (#) f2) ) thus ( g < r & g in dom (f1 (#) f2) ) by A15, VALUED_1:def_4; ::_thesis: verum end; hence f1 (#) f2 is divergent_in-infty_to+infty by A9, Def10; ::_thesis: verum end; theorem :: LIMFUNC1:54 for f1, f2 being PartFunc of REAL,REAL st f1 is divergent_in+infty_to+infty & ( for r being Real ex g being Real st ( r < g & g in dom (f1 + f2) ) ) & ex r being Real st f2 | (right_open_halfline r) is bounded_below holds f1 + f2 is divergent_in+infty_to+infty proof let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is divergent_in+infty_to+infty & ( for r being Real ex g being Real st ( r < g & g in dom (f1 + f2) ) ) & ex r being Real st f2 | (right_open_halfline r) is bounded_below implies f1 + f2 is divergent_in+infty_to+infty ) assume that A1: f1 is divergent_in+infty_to+infty and A2: for r being Real ex g being Real st ( r < g & g in dom (f1 + f2) ) ; ::_thesis: ( for r being Real holds not f2 | (right_open_halfline r) is bounded_below or f1 + f2 is divergent_in+infty_to+infty ) given r1 being Real such that A3: f2 | (right_open_halfline r1) is bounded_below ; ::_thesis: f1 + f2 is divergent_in+infty_to+infty now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_divergent_to+infty_&_rng_seq_c=_dom_(f1_+_f2)_holds_ (f1_+_f2)_/*_seq_is_divergent_to+infty let seq be Real_Sequence; ::_thesis: ( seq is divergent_to+infty & rng seq c= dom (f1 + f2) implies (f1 + f2) /* seq is divergent_to+infty ) assume that A4: seq is divergent_to+infty and A5: rng seq c= dom (f1 + f2) ; ::_thesis: (f1 + f2) /* seq is divergent_to+infty consider k being Element of NAT such that A6: for n being Element of NAT st k <= n holds r1 < seq . n by A4, Def4; A7: rng (seq ^\ k) c= rng seq by VALUED_0:21; dom (f1 + f2) = (dom f1) /\ (dom f2) by A5, Lm2; then rng (seq ^\ k) c= (dom f1) /\ (dom f2) by A5, A7, XBOOLE_1:1; then A8: (f1 /* (seq ^\ k)) + (f2 /* (seq ^\ k)) = (f1 + f2) /* (seq ^\ k) by RFUNCT_2:8 .= ((f1 + f2) /* seq) ^\ k by A5, VALUED_0:27 ; consider r2 being real number such that A9: for g being set st g in (right_open_halfline r1) /\ (dom f2) holds r2 <= f2 . g by A3, RFUNCT_1:71; A10: rng seq c= dom f2 by A5, Lm2; then A11: rng (seq ^\ k) c= dom f2 by A7, XBOOLE_1:1; now__::_thesis:_for_n_being_Element_of_NAT_holds_(-_(abs_r2))_-_1_<_(f2_/*_(seq_^\_k))_._n let n be Element of NAT ; ::_thesis: (- (abs r2)) - 1 < (f2 /* (seq ^\ k)) . n r1 < seq . (n + k) by A6, NAT_1:12; then ( (seq ^\ k) . n < +infty & r1 < (seq ^\ k) . n ) by NAT_1:def_3, XXREAL_0:9; then ( (seq ^\ k) . n in rng (seq ^\ k) & (seq ^\ k) . n in right_open_halfline r1 ) by VALUED_0:28, XXREAL_1:4; then (seq ^\ k) . n in (right_open_halfline r1) /\ (dom f2) by A11, XBOOLE_0:def_4; then r2 <= f2 . ((seq ^\ k) . n) by A9; then A12: r2 <= (f2 /* (seq ^\ k)) . n by A10, A7, FUNCT_2:108, XBOOLE_1:1; - (abs r2) <= r2 by ABSVALUE:4; then (- (abs r2)) - 1 < r2 - 0 by XREAL_1:15; hence (- (abs r2)) - 1 < (f2 /* (seq ^\ k)) . n by A12, XXREAL_0:2; ::_thesis: verum end; then A13: f2 /* (seq ^\ k) is bounded_below by SEQ_2:def_4; rng seq c= dom f1 by A5, Lm2; then A14: rng (seq ^\ k) c= dom f1 by A7, XBOOLE_1:1; seq ^\ k is divergent_to+infty by A4, Th26; then f1 /* (seq ^\ k) is divergent_to+infty by A1, A14, Def7; then (f1 /* (seq ^\ k)) + (f2 /* (seq ^\ k)) is divergent_to+infty by A13, Th9; hence (f1 + f2) /* seq is divergent_to+infty by A8, Th7; ::_thesis: verum end; hence f1 + f2 is divergent_in+infty_to+infty by A2, Def7; ::_thesis: verum end; theorem :: LIMFUNC1:55 for f1, f2 being PartFunc of REAL,REAL st f1 is divergent_in+infty_to+infty & ( for r being Real ex g being Real st ( r < g & g in dom (f1 (#) f2) ) ) & ex r, r1 being Real st ( 0 < r & ( for g being Real st g in (dom f2) /\ (right_open_halfline r1) holds r <= f2 . g ) ) holds f1 (#) f2 is divergent_in+infty_to+infty proof let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is divergent_in+infty_to+infty & ( for r being Real ex g being Real st ( r < g & g in dom (f1 (#) f2) ) ) & ex r, r1 being Real st ( 0 < r & ( for g being Real st g in (dom f2) /\ (right_open_halfline r1) holds r <= f2 . g ) ) implies f1 (#) f2 is divergent_in+infty_to+infty ) assume that A1: f1 is divergent_in+infty_to+infty and A2: for r being Real ex g being Real st ( r < g & g in dom (f1 (#) f2) ) ; ::_thesis: ( for r, r1 being Real holds ( not 0 < r or ex g being Real st ( g in (dom f2) /\ (right_open_halfline r1) & not r <= f2 . g ) ) or f1 (#) f2 is divergent_in+infty_to+infty ) given r2, r1 being Real such that A3: 0 < r2 and A4: for g being Real st g in (dom f2) /\ (right_open_halfline r1) holds r2 <= f2 . g ; ::_thesis: f1 (#) f2 is divergent_in+infty_to+infty now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_divergent_to+infty_&_rng_seq_c=_dom_(f1_(#)_f2)_holds_ (f1_(#)_f2)_/*_seq_is_divergent_to+infty let seq be Real_Sequence; ::_thesis: ( seq is divergent_to+infty & rng seq c= dom (f1 (#) f2) implies (f1 (#) f2) /* seq is divergent_to+infty ) assume that A5: seq is divergent_to+infty and A6: rng seq c= dom (f1 (#) f2) ; ::_thesis: (f1 (#) f2) /* seq is divergent_to+infty consider k being Element of NAT such that A7: for n being Element of NAT st k <= n holds r1 < seq . n by A5, Def4; A8: rng (seq ^\ k) c= rng seq by VALUED_0:21; A9: rng seq c= dom f2 by A6, Lm3; then A10: rng (seq ^\ k) c= dom f2 by A8, XBOOLE_1:1; A11: now__::_thesis:_(_0_<_r2_&_(_for_n_being_Element_of_NAT_holds_r2_<=_(f2_/*_(seq_^\_k))_._n_)_) thus 0 < r2 by A3; ::_thesis: for n being Element of NAT holds r2 <= (f2 /* (seq ^\ k)) . n let n be Element of NAT ; ::_thesis: r2 <= (f2 /* (seq ^\ k)) . n r1 < seq . (n + k) by A7, NAT_1:12; then r1 < (seq ^\ k) . n by NAT_1:def_3; then (seq ^\ k) . n in { g2 where g2 is Real : r1 < g2 } ; then ( (seq ^\ k) . n in rng (seq ^\ k) & (seq ^\ k) . n in right_open_halfline r1 ) by VALUED_0:28, XXREAL_1:230; then (seq ^\ k) . n in (dom f2) /\ (right_open_halfline r1) by A10, XBOOLE_0:def_4; then r2 <= f2 . ((seq ^\ k) . n) by A4; hence r2 <= (f2 /* (seq ^\ k)) . n by A9, A8, FUNCT_2:108, XBOOLE_1:1; ::_thesis: verum end; dom (f1 (#) f2) = (dom f1) /\ (dom f2) by A6, Lm3; then rng (seq ^\ k) c= (dom f1) /\ (dom f2) by A6, A8, XBOOLE_1:1; then A12: (f1 /* (seq ^\ k)) (#) (f2 /* (seq ^\ k)) = (f1 (#) f2) /* (seq ^\ k) by RFUNCT_2:8 .= ((f1 (#) f2) /* seq) ^\ k by A6, VALUED_0:27 ; rng seq c= dom f1 by A6, Lm3; then A13: rng (seq ^\ k) c= dom f1 by A8, XBOOLE_1:1; seq ^\ k is divergent_to+infty by A5, Th26; then f1 /* (seq ^\ k) is divergent_to+infty by A1, A13, Def7; then (f1 /* (seq ^\ k)) (#) (f2 /* (seq ^\ k)) is divergent_to+infty by A11, Th22; hence (f1 (#) f2) /* seq is divergent_to+infty by A12, Th7; ::_thesis: verum end; hence f1 (#) f2 is divergent_in+infty_to+infty by A2, Def7; ::_thesis: verum end; theorem :: LIMFUNC1:56 for f1, f2 being PartFunc of REAL,REAL st f1 is divergent_in-infty_to+infty & ( for r being Real ex g being Real st ( g < r & g in dom (f1 + f2) ) ) & ex r being Real st f2 | (left_open_halfline r) is bounded_below holds f1 + f2 is divergent_in-infty_to+infty proof let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is divergent_in-infty_to+infty & ( for r being Real ex g being Real st ( g < r & g in dom (f1 + f2) ) ) & ex r being Real st f2 | (left_open_halfline r) is bounded_below implies f1 + f2 is divergent_in-infty_to+infty ) assume that A1: f1 is divergent_in-infty_to+infty and A2: for r being Real ex g being Real st ( g < r & g in dom (f1 + f2) ) ; ::_thesis: ( for r being Real holds not f2 | (left_open_halfline r) is bounded_below or f1 + f2 is divergent_in-infty_to+infty ) given r1 being Real such that A3: f2 | (left_open_halfline r1) is bounded_below ; ::_thesis: f1 + f2 is divergent_in-infty_to+infty now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_divergent_to-infty_&_rng_seq_c=_dom_(f1_+_f2)_holds_ (f1_+_f2)_/*_seq_is_divergent_to+infty let seq be Real_Sequence; ::_thesis: ( seq is divergent_to-infty & rng seq c= dom (f1 + f2) implies (f1 + f2) /* seq is divergent_to+infty ) assume that A4: seq is divergent_to-infty and A5: rng seq c= dom (f1 + f2) ; ::_thesis: (f1 + f2) /* seq is divergent_to+infty consider k being Element of NAT such that A6: for n being Element of NAT st k <= n holds seq . n < r1 by A4, Def5; A7: rng (seq ^\ k) c= rng seq by VALUED_0:21; dom (f1 + f2) = (dom f1) /\ (dom f2) by A5, Lm2; then rng (seq ^\ k) c= (dom f1) /\ (dom f2) by A5, A7, XBOOLE_1:1; then A8: (f1 /* (seq ^\ k)) + (f2 /* (seq ^\ k)) = (f1 + f2) /* (seq ^\ k) by RFUNCT_2:8 .= ((f1 + f2) /* seq) ^\ k by A5, VALUED_0:27 ; consider r2 being real number such that A9: for g being set st g in (left_open_halfline r1) /\ (dom f2) holds r2 <= f2 . g by A3, RFUNCT_1:71; A10: rng seq c= dom f2 by A5, Lm2; then A11: rng (seq ^\ k) c= dom f2 by A7, XBOOLE_1:1; now__::_thesis:_for_n_being_Element_of_NAT_holds_(-_(abs_r2))_-_1_<_(f2_/*_(seq_^\_k))_._n let n be Element of NAT ; ::_thesis: (- (abs r2)) - 1 < (f2 /* (seq ^\ k)) . n seq . (n + k) < r1 by A6, NAT_1:12; then (seq ^\ k) . n < r1 by NAT_1:def_3; then (seq ^\ k) . n in { g2 where g2 is Real : g2 < r1 } ; then ( (seq ^\ k) . n in rng (seq ^\ k) & (seq ^\ k) . n in left_open_halfline r1 ) by VALUED_0:28, XXREAL_1:229; then (seq ^\ k) . n in (left_open_halfline r1) /\ (dom f2) by A11, XBOOLE_0:def_4; then r2 <= f2 . ((seq ^\ k) . n) by A9; then A12: r2 <= (f2 /* (seq ^\ k)) . n by A10, A7, FUNCT_2:108, XBOOLE_1:1; - (abs r2) <= r2 by ABSVALUE:4; then (- (abs r2)) - 1 < r2 - 0 by XREAL_1:15; hence (- (abs r2)) - 1 < (f2 /* (seq ^\ k)) . n by A12, XXREAL_0:2; ::_thesis: verum end; then A13: f2 /* (seq ^\ k) is bounded_below by SEQ_2:def_4; rng seq c= dom f1 by A5, Lm2; then A14: rng (seq ^\ k) c= dom f1 by A7, XBOOLE_1:1; seq ^\ k is divergent_to-infty by A4, Th27; then f1 /* (seq ^\ k) is divergent_to+infty by A1, A14, Def10; then (f1 /* (seq ^\ k)) + (f2 /* (seq ^\ k)) is divergent_to+infty by A13, Th9; hence (f1 + f2) /* seq is divergent_to+infty by A8, Th7; ::_thesis: verum end; hence f1 + f2 is divergent_in-infty_to+infty by A2, Def10; ::_thesis: verum end; theorem :: LIMFUNC1:57 for f1, f2 being PartFunc of REAL,REAL st f1 is divergent_in-infty_to+infty & ( for r being Real ex g being Real st ( g < r & g in dom (f1 (#) f2) ) ) & ex r, r1 being Real st ( 0 < r & ( for g being Real st g in (dom f2) /\ (left_open_halfline r1) holds r <= f2 . g ) ) holds f1 (#) f2 is divergent_in-infty_to+infty proof let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is divergent_in-infty_to+infty & ( for r being Real ex g being Real st ( g < r & g in dom (f1 (#) f2) ) ) & ex r, r1 being Real st ( 0 < r & ( for g being Real st g in (dom f2) /\ (left_open_halfline r1) holds r <= f2 . g ) ) implies f1 (#) f2 is divergent_in-infty_to+infty ) assume that A1: f1 is divergent_in-infty_to+infty and A2: for r being Real ex g being Real st ( g < r & g in dom (f1 (#) f2) ) ; ::_thesis: ( for r, r1 being Real holds ( not 0 < r or ex g being Real st ( g in (dom f2) /\ (left_open_halfline r1) & not r <= f2 . g ) ) or f1 (#) f2 is divergent_in-infty_to+infty ) given r2, r1 being Real such that A3: 0 < r2 and A4: for g being Real st g in (dom f2) /\ (left_open_halfline r1) holds r2 <= f2 . g ; ::_thesis: f1 (#) f2 is divergent_in-infty_to+infty now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_divergent_to-infty_&_rng_seq_c=_dom_(f1_(#)_f2)_holds_ (f1_(#)_f2)_/*_seq_is_divergent_to+infty let seq be Real_Sequence; ::_thesis: ( seq is divergent_to-infty & rng seq c= dom (f1 (#) f2) implies (f1 (#) f2) /* seq is divergent_to+infty ) assume that A5: seq is divergent_to-infty and A6: rng seq c= dom (f1 (#) f2) ; ::_thesis: (f1 (#) f2) /* seq is divergent_to+infty consider k being Element of NAT such that A7: for n being Element of NAT st k <= n holds seq . n < r1 by A5, Def5; A8: rng (seq ^\ k) c= rng seq by VALUED_0:21; A9: rng seq c= dom f2 by A6, Lm3; then A10: rng (seq ^\ k) c= dom f2 by A8, XBOOLE_1:1; A11: now__::_thesis:_(_0_<_r2_&_(_for_n_being_Element_of_NAT_holds_r2_<=_(f2_/*_(seq_^\_k))_._n_)_) thus 0 < r2 by A3; ::_thesis: for n being Element of NAT holds r2 <= (f2 /* (seq ^\ k)) . n let n be Element of NAT ; ::_thesis: r2 <= (f2 /* (seq ^\ k)) . n seq . (n + k) < r1 by A7, NAT_1:12; then (seq ^\ k) . n < r1 by NAT_1:def_3; then (seq ^\ k) . n in { g2 where g2 is Real : g2 < r1 } ; then ( (seq ^\ k) . n in rng (seq ^\ k) & (seq ^\ k) . n in left_open_halfline r1 ) by VALUED_0:28, XXREAL_1:229; then (seq ^\ k) . n in (dom f2) /\ (left_open_halfline r1) by A10, XBOOLE_0:def_4; then r2 <= f2 . ((seq ^\ k) . n) by A4; hence r2 <= (f2 /* (seq ^\ k)) . n by A9, A8, FUNCT_2:108, XBOOLE_1:1; ::_thesis: verum end; dom (f1 (#) f2) = (dom f1) /\ (dom f2) by A6, Lm3; then rng (seq ^\ k) c= (dom f1) /\ (dom f2) by A6, A8, XBOOLE_1:1; then A12: (f1 /* (seq ^\ k)) (#) (f2 /* (seq ^\ k)) = (f1 (#) f2) /* (seq ^\ k) by RFUNCT_2:8 .= ((f1 (#) f2) /* seq) ^\ k by A6, VALUED_0:27 ; rng seq c= dom f1 by A6, Lm3; then A13: rng (seq ^\ k) c= dom f1 by A8, XBOOLE_1:1; seq ^\ k is divergent_to-infty by A5, Th27; then f1 /* (seq ^\ k) is divergent_to+infty by A1, A13, Def10; then (f1 /* (seq ^\ k)) (#) (f2 /* (seq ^\ k)) is divergent_to+infty by A11, Th22; hence (f1 (#) f2) /* seq is divergent_to+infty by A12, Th7; ::_thesis: verum end; hence f1 (#) f2 is divergent_in-infty_to+infty by A2, Def10; ::_thesis: verum end; theorem :: LIMFUNC1:58 for f being PartFunc of REAL,REAL for r being Real holds ( ( f is divergent_in+infty_to+infty & r > 0 implies r (#) f is divergent_in+infty_to+infty ) & ( f is divergent_in+infty_to+infty & r < 0 implies r (#) f is divergent_in+infty_to-infty ) & ( f is divergent_in+infty_to-infty & r > 0 implies r (#) f is divergent_in+infty_to-infty ) & ( f is divergent_in+infty_to-infty & r < 0 implies r (#) f is divergent_in+infty_to+infty ) ) proof let f be PartFunc of REAL,REAL; ::_thesis: for r being Real holds ( ( f is divergent_in+infty_to+infty & r > 0 implies r (#) f is divergent_in+infty_to+infty ) & ( f is divergent_in+infty_to+infty & r < 0 implies r (#) f is divergent_in+infty_to-infty ) & ( f is divergent_in+infty_to-infty & r > 0 implies r (#) f is divergent_in+infty_to-infty ) & ( f is divergent_in+infty_to-infty & r < 0 implies r (#) f is divergent_in+infty_to+infty ) ) let r be Real; ::_thesis: ( ( f is divergent_in+infty_to+infty & r > 0 implies r (#) f is divergent_in+infty_to+infty ) & ( f is divergent_in+infty_to+infty & r < 0 implies r (#) f is divergent_in+infty_to-infty ) & ( f is divergent_in+infty_to-infty & r > 0 implies r (#) f is divergent_in+infty_to-infty ) & ( f is divergent_in+infty_to-infty & r < 0 implies r (#) f is divergent_in+infty_to+infty ) ) thus ( f is divergent_in+infty_to+infty & r > 0 implies r (#) f is divergent_in+infty_to+infty ) ::_thesis: ( ( f is divergent_in+infty_to+infty & r < 0 implies r (#) f is divergent_in+infty_to-infty ) & ( f is divergent_in+infty_to-infty & r > 0 implies r (#) f is divergent_in+infty_to-infty ) & ( f is divergent_in+infty_to-infty & r < 0 implies r (#) f is divergent_in+infty_to+infty ) ) proof assume that A1: f is divergent_in+infty_to+infty and A2: r > 0 ; ::_thesis: r (#) f is divergent_in+infty_to+infty A3: now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_divergent_to+infty_&_rng_seq_c=_dom_(r_(#)_f)_holds_ (r_(#)_f)_/*_seq_is_divergent_to+infty let seq be Real_Sequence; ::_thesis: ( seq is divergent_to+infty & rng seq c= dom (r (#) f) implies (r (#) f) /* seq is divergent_to+infty ) assume that A4: seq is divergent_to+infty and A5: rng seq c= dom (r (#) f) ; ::_thesis: (r (#) f) /* seq is divergent_to+infty A6: rng seq c= dom f by A5, VALUED_1:def_5; then f /* seq is divergent_to+infty by A1, A4, Def7; then r (#) (f /* seq) is divergent_to+infty by A2, Th13; hence (r (#) f) /* seq is divergent_to+infty by A6, RFUNCT_2:9; ::_thesis: verum end; now__::_thesis:_for_r1_being_Real_ex_g_being_Real_st_ (_r1_<_g_&_g_in_dom_(r_(#)_f)_) let r1 be Real; ::_thesis: ex g being Real st ( r1 < g & g in dom (r (#) f) ) consider g being Real such that A7: ( r1 < g & g in dom f ) by A1, Def7; take g = g; ::_thesis: ( r1 < g & g in dom (r (#) f) ) thus ( r1 < g & g in dom (r (#) f) ) by A7, VALUED_1:def_5; ::_thesis: verum end; hence r (#) f is divergent_in+infty_to+infty by A3, Def7; ::_thesis: verum end; thus ( f is divergent_in+infty_to+infty & r < 0 implies r (#) f is divergent_in+infty_to-infty ) ::_thesis: ( ( f is divergent_in+infty_to-infty & r > 0 implies r (#) f is divergent_in+infty_to-infty ) & ( f is divergent_in+infty_to-infty & r < 0 implies r (#) f is divergent_in+infty_to+infty ) ) proof assume that A8: f is divergent_in+infty_to+infty and A9: r < 0 ; ::_thesis: r (#) f is divergent_in+infty_to-infty A10: now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_divergent_to+infty_&_rng_seq_c=_dom_(r_(#)_f)_holds_ (r_(#)_f)_/*_seq_is_divergent_to-infty let seq be Real_Sequence; ::_thesis: ( seq is divergent_to+infty & rng seq c= dom (r (#) f) implies (r (#) f) /* seq is divergent_to-infty ) assume that A11: seq is divergent_to+infty and A12: rng seq c= dom (r (#) f) ; ::_thesis: (r (#) f) /* seq is divergent_to-infty A13: rng seq c= dom f by A12, VALUED_1:def_5; then f /* seq is divergent_to+infty by A8, A11, Def7; then r (#) (f /* seq) is divergent_to-infty by A9, Th13; hence (r (#) f) /* seq is divergent_to-infty by A13, RFUNCT_2:9; ::_thesis: verum end; now__::_thesis:_for_r1_being_Real_ex_g_being_Real_st_ (_r1_<_g_&_g_in_dom_(r_(#)_f)_) let r1 be Real; ::_thesis: ex g being Real st ( r1 < g & g in dom (r (#) f) ) consider g being Real such that A14: ( r1 < g & g in dom f ) by A8, Def7; take g = g; ::_thesis: ( r1 < g & g in dom (r (#) f) ) thus ( r1 < g & g in dom (r (#) f) ) by A14, VALUED_1:def_5; ::_thesis: verum end; hence r (#) f is divergent_in+infty_to-infty by A10, Def8; ::_thesis: verum end; thus ( f is divergent_in+infty_to-infty & r > 0 implies r (#) f is divergent_in+infty_to-infty ) ::_thesis: ( f is divergent_in+infty_to-infty & r < 0 implies r (#) f is divergent_in+infty_to+infty ) proof assume that A15: f is divergent_in+infty_to-infty and A16: r > 0 ; ::_thesis: r (#) f is divergent_in+infty_to-infty A17: now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_divergent_to+infty_&_rng_seq_c=_dom_(r_(#)_f)_holds_ (r_(#)_f)_/*_seq_is_divergent_to-infty let seq be Real_Sequence; ::_thesis: ( seq is divergent_to+infty & rng seq c= dom (r (#) f) implies (r (#) f) /* seq is divergent_to-infty ) assume that A18: seq is divergent_to+infty and A19: rng seq c= dom (r (#) f) ; ::_thesis: (r (#) f) /* seq is divergent_to-infty A20: rng seq c= dom f by A19, VALUED_1:def_5; then f /* seq is divergent_to-infty by A15, A18, Def8; then r (#) (f /* seq) is divergent_to-infty by A16, Th14; hence (r (#) f) /* seq is divergent_to-infty by A20, RFUNCT_2:9; ::_thesis: verum end; now__::_thesis:_for_r1_being_Real_ex_g_being_Real_st_ (_r1_<_g_&_g_in_dom_(r_(#)_f)_) let r1 be Real; ::_thesis: ex g being Real st ( r1 < g & g in dom (r (#) f) ) consider g being Real such that A21: ( r1 < g & g in dom f ) by A15, Def8; take g = g; ::_thesis: ( r1 < g & g in dom (r (#) f) ) thus ( r1 < g & g in dom (r (#) f) ) by A21, VALUED_1:def_5; ::_thesis: verum end; hence r (#) f is divergent_in+infty_to-infty by A17, Def8; ::_thesis: verum end; assume that A22: f is divergent_in+infty_to-infty and A23: r < 0 ; ::_thesis: r (#) f is divergent_in+infty_to+infty A24: now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_divergent_to+infty_&_rng_seq_c=_dom_(r_(#)_f)_holds_ (r_(#)_f)_/*_seq_is_divergent_to+infty let seq be Real_Sequence; ::_thesis: ( seq is divergent_to+infty & rng seq c= dom (r (#) f) implies (r (#) f) /* seq is divergent_to+infty ) assume that A25: seq is divergent_to+infty and A26: rng seq c= dom (r (#) f) ; ::_thesis: (r (#) f) /* seq is divergent_to+infty A27: rng seq c= dom f by A26, VALUED_1:def_5; then f /* seq is divergent_to-infty by A22, A25, Def8; then r (#) (f /* seq) is divergent_to+infty by A23, Th14; hence (r (#) f) /* seq is divergent_to+infty by A27, RFUNCT_2:9; ::_thesis: verum end; now__::_thesis:_for_r1_being_Real_ex_g_being_Real_st_ (_r1_<_g_&_g_in_dom_(r_(#)_f)_) let r1 be Real; ::_thesis: ex g being Real st ( r1 < g & g in dom (r (#) f) ) consider g being Real such that A28: ( r1 < g & g in dom f ) by A22, Def8; take g = g; ::_thesis: ( r1 < g & g in dom (r (#) f) ) thus ( r1 < g & g in dom (r (#) f) ) by A28, VALUED_1:def_5; ::_thesis: verum end; hence r (#) f is divergent_in+infty_to+infty by A24, Def7; ::_thesis: verum end; theorem :: LIMFUNC1:59 for f being PartFunc of REAL,REAL for r being Real holds ( ( f is divergent_in-infty_to+infty & r > 0 implies r (#) f is divergent_in-infty_to+infty ) & ( f is divergent_in-infty_to+infty & r < 0 implies r (#) f is divergent_in-infty_to-infty ) & ( f is divergent_in-infty_to-infty & r > 0 implies r (#) f is divergent_in-infty_to-infty ) & ( f is divergent_in-infty_to-infty & r < 0 implies r (#) f is divergent_in-infty_to+infty ) ) proof let f be PartFunc of REAL,REAL; ::_thesis: for r being Real holds ( ( f is divergent_in-infty_to+infty & r > 0 implies r (#) f is divergent_in-infty_to+infty ) & ( f is divergent_in-infty_to+infty & r < 0 implies r (#) f is divergent_in-infty_to-infty ) & ( f is divergent_in-infty_to-infty & r > 0 implies r (#) f is divergent_in-infty_to-infty ) & ( f is divergent_in-infty_to-infty & r < 0 implies r (#) f is divergent_in-infty_to+infty ) ) let r be Real; ::_thesis: ( ( f is divergent_in-infty_to+infty & r > 0 implies r (#) f is divergent_in-infty_to+infty ) & ( f is divergent_in-infty_to+infty & r < 0 implies r (#) f is divergent_in-infty_to-infty ) & ( f is divergent_in-infty_to-infty & r > 0 implies r (#) f is divergent_in-infty_to-infty ) & ( f is divergent_in-infty_to-infty & r < 0 implies r (#) f is divergent_in-infty_to+infty ) ) thus ( f is divergent_in-infty_to+infty & r > 0 implies r (#) f is divergent_in-infty_to+infty ) ::_thesis: ( ( f is divergent_in-infty_to+infty & r < 0 implies r (#) f is divergent_in-infty_to-infty ) & ( f is divergent_in-infty_to-infty & r > 0 implies r (#) f is divergent_in-infty_to-infty ) & ( f is divergent_in-infty_to-infty & r < 0 implies r (#) f is divergent_in-infty_to+infty ) ) proof assume that A1: f is divergent_in-infty_to+infty and A2: r > 0 ; ::_thesis: r (#) f is divergent_in-infty_to+infty A3: now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_divergent_to-infty_&_rng_seq_c=_dom_(r_(#)_f)_holds_ (r_(#)_f)_/*_seq_is_divergent_to+infty let seq be Real_Sequence; ::_thesis: ( seq is divergent_to-infty & rng seq c= dom (r (#) f) implies (r (#) f) /* seq is divergent_to+infty ) assume that A4: seq is divergent_to-infty and A5: rng seq c= dom (r (#) f) ; ::_thesis: (r (#) f) /* seq is divergent_to+infty A6: rng seq c= dom f by A5, VALUED_1:def_5; then f /* seq is divergent_to+infty by A1, A4, Def10; then r (#) (f /* seq) is divergent_to+infty by A2, Th13; hence (r (#) f) /* seq is divergent_to+infty by A6, RFUNCT_2:9; ::_thesis: verum end; now__::_thesis:_for_r1_being_Real_ex_g_being_Real_st_ (_g_<_r1_&_g_in_dom_(r_(#)_f)_) let r1 be Real; ::_thesis: ex g being Real st ( g < r1 & g in dom (r (#) f) ) consider g being Real such that A7: ( g < r1 & g in dom f ) by A1, Def10; take g = g; ::_thesis: ( g < r1 & g in dom (r (#) f) ) thus ( g < r1 & g in dom (r (#) f) ) by A7, VALUED_1:def_5; ::_thesis: verum end; hence r (#) f is divergent_in-infty_to+infty by A3, Def10; ::_thesis: verum end; thus ( f is divergent_in-infty_to+infty & r < 0 implies r (#) f is divergent_in-infty_to-infty ) ::_thesis: ( ( f is divergent_in-infty_to-infty & r > 0 implies r (#) f is divergent_in-infty_to-infty ) & ( f is divergent_in-infty_to-infty & r < 0 implies r (#) f is divergent_in-infty_to+infty ) ) proof assume that A8: f is divergent_in-infty_to+infty and A9: r < 0 ; ::_thesis: r (#) f is divergent_in-infty_to-infty A10: now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_divergent_to-infty_&_rng_seq_c=_dom_(r_(#)_f)_holds_ (r_(#)_f)_/*_seq_is_divergent_to-infty let seq be Real_Sequence; ::_thesis: ( seq is divergent_to-infty & rng seq c= dom (r (#) f) implies (r (#) f) /* seq is divergent_to-infty ) assume that A11: seq is divergent_to-infty and A12: rng seq c= dom (r (#) f) ; ::_thesis: (r (#) f) /* seq is divergent_to-infty A13: rng seq c= dom f by A12, VALUED_1:def_5; then f /* seq is divergent_to+infty by A8, A11, Def10; then r (#) (f /* seq) is divergent_to-infty by A9, Th13; hence (r (#) f) /* seq is divergent_to-infty by A13, RFUNCT_2:9; ::_thesis: verum end; now__::_thesis:_for_r1_being_Real_ex_g_being_Real_st_ (_g_<_r1_&_g_in_dom_(r_(#)_f)_) let r1 be Real; ::_thesis: ex g being Real st ( g < r1 & g in dom (r (#) f) ) consider g being Real such that A14: ( g < r1 & g in dom f ) by A8, Def10; take g = g; ::_thesis: ( g < r1 & g in dom (r (#) f) ) thus ( g < r1 & g in dom (r (#) f) ) by A14, VALUED_1:def_5; ::_thesis: verum end; hence r (#) f is divergent_in-infty_to-infty by A10, Def11; ::_thesis: verum end; thus ( f is divergent_in-infty_to-infty & r > 0 implies r (#) f is divergent_in-infty_to-infty ) ::_thesis: ( f is divergent_in-infty_to-infty & r < 0 implies r (#) f is divergent_in-infty_to+infty ) proof assume that A15: f is divergent_in-infty_to-infty and A16: r > 0 ; ::_thesis: r (#) f is divergent_in-infty_to-infty A17: now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_divergent_to-infty_&_rng_seq_c=_dom_(r_(#)_f)_holds_ (r_(#)_f)_/*_seq_is_divergent_to-infty let seq be Real_Sequence; ::_thesis: ( seq is divergent_to-infty & rng seq c= dom (r (#) f) implies (r (#) f) /* seq is divergent_to-infty ) assume that A18: seq is divergent_to-infty and A19: rng seq c= dom (r (#) f) ; ::_thesis: (r (#) f) /* seq is divergent_to-infty A20: rng seq c= dom f by A19, VALUED_1:def_5; then f /* seq is divergent_to-infty by A15, A18, Def11; then r (#) (f /* seq) is divergent_to-infty by A16, Th14; hence (r (#) f) /* seq is divergent_to-infty by A20, RFUNCT_2:9; ::_thesis: verum end; now__::_thesis:_for_r1_being_Real_ex_g_being_Real_st_ (_g_<_r1_&_g_in_dom_(r_(#)_f)_) let r1 be Real; ::_thesis: ex g being Real st ( g < r1 & g in dom (r (#) f) ) consider g being Real such that A21: ( g < r1 & g in dom f ) by A15, Def11; take g = g; ::_thesis: ( g < r1 & g in dom (r (#) f) ) thus ( g < r1 & g in dom (r (#) f) ) by A21, VALUED_1:def_5; ::_thesis: verum end; hence r (#) f is divergent_in-infty_to-infty by A17, Def11; ::_thesis: verum end; assume that A22: f is divergent_in-infty_to-infty and A23: r < 0 ; ::_thesis: r (#) f is divergent_in-infty_to+infty A24: now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_divergent_to-infty_&_rng_seq_c=_dom_(r_(#)_f)_holds_ (r_(#)_f)_/*_seq_is_divergent_to+infty let seq be Real_Sequence; ::_thesis: ( seq is divergent_to-infty & rng seq c= dom (r (#) f) implies (r (#) f) /* seq is divergent_to+infty ) assume that A25: seq is divergent_to-infty and A26: rng seq c= dom (r (#) f) ; ::_thesis: (r (#) f) /* seq is divergent_to+infty A27: rng seq c= dom f by A26, VALUED_1:def_5; then f /* seq is divergent_to-infty by A22, A25, Def11; then r (#) (f /* seq) is divergent_to+infty by A23, Th14; hence (r (#) f) /* seq is divergent_to+infty by A27, RFUNCT_2:9; ::_thesis: verum end; now__::_thesis:_for_r1_being_Real_ex_g_being_Real_st_ (_g_<_r1_&_g_in_dom_(r_(#)_f)_) let r1 be Real; ::_thesis: ex g being Real st ( g < r1 & g in dom (r (#) f) ) consider g being Real such that A28: ( g < r1 & g in dom f ) by A22, Def11; take g = g; ::_thesis: ( g < r1 & g in dom (r (#) f) ) thus ( g < r1 & g in dom (r (#) f) ) by A28, VALUED_1:def_5; ::_thesis: verum end; hence r (#) f is divergent_in-infty_to+infty by A24, Def10; ::_thesis: verum end; theorem :: LIMFUNC1:60 for f being PartFunc of REAL,REAL st ( f is divergent_in+infty_to+infty or f is divergent_in+infty_to-infty ) holds abs f is divergent_in+infty_to+infty proof let f be PartFunc of REAL,REAL; ::_thesis: ( ( f is divergent_in+infty_to+infty or f is divergent_in+infty_to-infty ) implies abs f is divergent_in+infty_to+infty ) assume A1: ( f is divergent_in+infty_to+infty or f is divergent_in+infty_to-infty ) ; ::_thesis: abs f is divergent_in+infty_to+infty now__::_thesis:_abs_f_is_divergent_in+infty_to+infty percases ( f is divergent_in+infty_to+infty or f is divergent_in+infty_to-infty ) by A1; supposeA2: f is divergent_in+infty_to+infty ; ::_thesis: abs f is divergent_in+infty_to+infty A3: now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_divergent_to+infty_&_rng_seq_c=_dom_(abs_f)_holds_ (abs_f)_/*_seq_is_divergent_to+infty let seq be Real_Sequence; ::_thesis: ( seq is divergent_to+infty & rng seq c= dom (abs f) implies (abs f) /* seq is divergent_to+infty ) assume that A4: seq is divergent_to+infty and A5: rng seq c= dom (abs f) ; ::_thesis: (abs f) /* seq is divergent_to+infty A6: rng seq c= dom f by A5, VALUED_1:def_11; then f /* seq is divergent_to+infty by A2, A4, Def7; then abs (f /* seq) is divergent_to+infty by Th25; hence (abs f) /* seq is divergent_to+infty by A6, RFUNCT_2:10; ::_thesis: verum end; now__::_thesis:_for_r_being_Real_ex_g_being_Real_st_ (_r_<_g_&_g_in_dom_(abs_f)_) let r be Real; ::_thesis: ex g being Real st ( r < g & g in dom (abs f) ) consider g being Real such that A7: ( r < g & g in dom f ) by A2, Def7; take g = g; ::_thesis: ( r < g & g in dom (abs f) ) thus ( r < g & g in dom (abs f) ) by A7, VALUED_1:def_11; ::_thesis: verum end; hence abs f is divergent_in+infty_to+infty by A3, Def7; ::_thesis: verum end; supposeA8: f is divergent_in+infty_to-infty ; ::_thesis: abs f is divergent_in+infty_to+infty A9: now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_divergent_to+infty_&_rng_seq_c=_dom_(abs_f)_holds_ (abs_f)_/*_seq_is_divergent_to+infty let seq be Real_Sequence; ::_thesis: ( seq is divergent_to+infty & rng seq c= dom (abs f) implies (abs f) /* seq is divergent_to+infty ) assume that A10: seq is divergent_to+infty and A11: rng seq c= dom (abs f) ; ::_thesis: (abs f) /* seq is divergent_to+infty A12: rng seq c= dom f by A11, VALUED_1:def_11; then f /* seq is divergent_to-infty by A8, A10, Def8; then abs (f /* seq) is divergent_to+infty by Th25; hence (abs f) /* seq is divergent_to+infty by A12, RFUNCT_2:10; ::_thesis: verum end; now__::_thesis:_for_r_being_Real_ex_g_being_Real_st_ (_r_<_g_&_g_in_dom_(abs_f)_) let r be Real; ::_thesis: ex g being Real st ( r < g & g in dom (abs f) ) consider g being Real such that A13: ( r < g & g in dom f ) by A8, Def8; take g = g; ::_thesis: ( r < g & g in dom (abs f) ) thus ( r < g & g in dom (abs f) ) by A13, VALUED_1:def_11; ::_thesis: verum end; hence abs f is divergent_in+infty_to+infty by A9, Def7; ::_thesis: verum end; end; end; hence abs f is divergent_in+infty_to+infty ; ::_thesis: verum end; theorem :: LIMFUNC1:61 for f being PartFunc of REAL,REAL st ( f is divergent_in-infty_to+infty or f is divergent_in-infty_to-infty ) holds abs f is divergent_in-infty_to+infty proof let f be PartFunc of REAL,REAL; ::_thesis: ( ( f is divergent_in-infty_to+infty or f is divergent_in-infty_to-infty ) implies abs f is divergent_in-infty_to+infty ) assume A1: ( f is divergent_in-infty_to+infty or f is divergent_in-infty_to-infty ) ; ::_thesis: abs f is divergent_in-infty_to+infty now__::_thesis:_abs_f_is_divergent_in-infty_to+infty percases ( f is divergent_in-infty_to+infty or f is divergent_in-infty_to-infty ) by A1; supposeA2: f is divergent_in-infty_to+infty ; ::_thesis: abs f is divergent_in-infty_to+infty A3: now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_divergent_to-infty_&_rng_seq_c=_dom_(abs_f)_holds_ (abs_f)_/*_seq_is_divergent_to+infty let seq be Real_Sequence; ::_thesis: ( seq is divergent_to-infty & rng seq c= dom (abs f) implies (abs f) /* seq is divergent_to+infty ) assume that A4: seq is divergent_to-infty and A5: rng seq c= dom (abs f) ; ::_thesis: (abs f) /* seq is divergent_to+infty A6: rng seq c= dom f by A5, VALUED_1:def_11; then f /* seq is divergent_to+infty by A2, A4, Def10; then abs (f /* seq) is divergent_to+infty by Th25; hence (abs f) /* seq is divergent_to+infty by A6, RFUNCT_2:10; ::_thesis: verum end; now__::_thesis:_for_r_being_Real_ex_g_being_Real_st_ (_g_<_r_&_g_in_dom_(abs_f)_) let r be Real; ::_thesis: ex g being Real st ( g < r & g in dom (abs f) ) consider g being Real such that A7: ( g < r & g in dom f ) by A2, Def10; take g = g; ::_thesis: ( g < r & g in dom (abs f) ) thus ( g < r & g in dom (abs f) ) by A7, VALUED_1:def_11; ::_thesis: verum end; hence abs f is divergent_in-infty_to+infty by A3, Def10; ::_thesis: verum end; supposeA8: f is divergent_in-infty_to-infty ; ::_thesis: abs f is divergent_in-infty_to+infty A9: now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_divergent_to-infty_&_rng_seq_c=_dom_(abs_f)_holds_ (abs_f)_/*_seq_is_divergent_to+infty let seq be Real_Sequence; ::_thesis: ( seq is divergent_to-infty & rng seq c= dom (abs f) implies (abs f) /* seq is divergent_to+infty ) assume that A10: seq is divergent_to-infty and A11: rng seq c= dom (abs f) ; ::_thesis: (abs f) /* seq is divergent_to+infty A12: rng seq c= dom f by A11, VALUED_1:def_11; then f /* seq is divergent_to-infty by A8, A10, Def11; then abs (f /* seq) is divergent_to+infty by Th25; hence (abs f) /* seq is divergent_to+infty by A12, RFUNCT_2:10; ::_thesis: verum end; now__::_thesis:_for_r_being_Real_ex_g_being_Real_st_ (_g_<_r_&_g_in_dom_(abs_f)_) let r be Real; ::_thesis: ex g being Real st ( g < r & g in dom (abs f) ) consider g being Real such that A13: ( g < r & g in dom f ) by A8, Def11; take g = g; ::_thesis: ( g < r & g in dom (abs f) ) thus ( g < r & g in dom (abs f) ) by A13, VALUED_1:def_11; ::_thesis: verum end; hence abs f is divergent_in-infty_to+infty by A9, Def10; ::_thesis: verum end; end; end; hence abs f is divergent_in-infty_to+infty ; ::_thesis: verum end; theorem Th62: :: LIMFUNC1:62 for f being PartFunc of REAL,REAL st ex r being Real st ( f | (right_open_halfline r) is non-decreasing & not f | (right_open_halfline r) is bounded_above ) & ( for r being Real ex g being Real st ( r < g & g in dom f ) ) holds f is divergent_in+infty_to+infty proof let f be PartFunc of REAL,REAL; ::_thesis: ( ex r being Real st ( f | (right_open_halfline r) is non-decreasing & not f | (right_open_halfline r) is bounded_above ) & ( for r being Real ex g being Real st ( r < g & g in dom f ) ) implies f is divergent_in+infty_to+infty ) given r1 being Real such that A1: f | (right_open_halfline r1) is non-decreasing and A2: not f | (right_open_halfline r1) is bounded_above ; ::_thesis: ( ex r being Real st for g being Real holds ( not r < g or not g in dom f ) or f is divergent_in+infty_to+infty ) A3: now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_divergent_to+infty_&_rng_seq_c=_dom_f_holds_ f_/*_seq_is_divergent_to+infty let seq be Real_Sequence; ::_thesis: ( seq is divergent_to+infty & rng seq c= dom f implies f /* seq is divergent_to+infty ) assume that A4: seq is divergent_to+infty and A5: rng seq c= dom f ; ::_thesis: f /* seq is divergent_to+infty now__::_thesis:_for_r_being_Real_ex_n_being_Element_of_NAT_st_ for_m_being_Element_of_NAT_st_n_<=_m_holds_ r_<_(f_/*_seq)_._m let r be Real; ::_thesis: ex n being Element of NAT st for m being Element of NAT st n <= m holds r < (f /* seq) . m consider g1 being set such that A6: g1 in (right_open_halfline r1) /\ (dom f) and A7: r < f . g1 by A2, RFUNCT_1:70; reconsider g1 = g1 as Real by A6; consider n being Element of NAT such that A8: for m being Element of NAT st n <= m holds (abs g1) + (abs r1) < seq . m by A4, Def4; take n = n; ::_thesis: for m being Element of NAT st n <= m holds r < (f /* seq) . m let m be Element of NAT ; ::_thesis: ( n <= m implies r < (f /* seq) . m ) assume n <= m ; ::_thesis: r < (f /* seq) . m then A9: (abs g1) + (abs r1) < seq . m by A8; ( r1 <= abs r1 & 0 <= abs g1 ) by ABSVALUE:4, COMPLEX1:46; then 0 + r1 <= (abs g1) + (abs r1) by XREAL_1:7; then r1 < seq . m by A9, XXREAL_0:2; then seq . m in { g2 where g2 is Real : r1 < g2 } ; then ( seq . m in rng seq & seq . m in right_open_halfline r1 ) by VALUED_0:28, XXREAL_1:230; then A10: seq . m in (right_open_halfline r1) /\ (dom f) by A5, XBOOLE_0:def_4; ( g1 <= abs g1 & 0 <= abs r1 ) by ABSVALUE:4, COMPLEX1:46; then g1 + 0 <= (abs g1) + (abs r1) by XREAL_1:7; then g1 < seq . m by A9, XXREAL_0:2; then f . g1 <= f . (seq . m) by A1, A6, A10, RFUNCT_2:22; then r < f . (seq . m) by A7, XXREAL_0:2; hence r < (f /* seq) . m by A5, FUNCT_2:108; ::_thesis: verum end; hence f /* seq is divergent_to+infty by Def4; ::_thesis: verum end; assume for r being Real ex g being Real st ( r < g & g in dom f ) ; ::_thesis: f is divergent_in+infty_to+infty hence f is divergent_in+infty_to+infty by A3, Def7; ::_thesis: verum end; theorem :: LIMFUNC1:63 for f being PartFunc of REAL,REAL st ex r being Real st ( f | (right_open_halfline r) is increasing & not f | (right_open_halfline r) is bounded_above ) & ( for r being Real ex g being Real st ( r < g & g in dom f ) ) holds f is divergent_in+infty_to+infty by Th62; theorem Th64: :: LIMFUNC1:64 for f being PartFunc of REAL,REAL st ex r being Real st ( f | (right_open_halfline r) is non-increasing & not f | (right_open_halfline r) is bounded_below ) & ( for r being Real ex g being Real st ( r < g & g in dom f ) ) holds f is divergent_in+infty_to-infty proof let f be PartFunc of REAL,REAL; ::_thesis: ( ex r being Real st ( f | (right_open_halfline r) is non-increasing & not f | (right_open_halfline r) is bounded_below ) & ( for r being Real ex g being Real st ( r < g & g in dom f ) ) implies f is divergent_in+infty_to-infty ) given r1 being Real such that A1: f | (right_open_halfline r1) is non-increasing and A2: not f | (right_open_halfline r1) is bounded_below ; ::_thesis: ( ex r being Real st for g being Real holds ( not r < g or not g in dom f ) or f is divergent_in+infty_to-infty ) A3: now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_divergent_to+infty_&_rng_seq_c=_dom_f_holds_ f_/*_seq_is_divergent_to-infty let seq be Real_Sequence; ::_thesis: ( seq is divergent_to+infty & rng seq c= dom f implies f /* seq is divergent_to-infty ) assume that A4: seq is divergent_to+infty and A5: rng seq c= dom f ; ::_thesis: f /* seq is divergent_to-infty now__::_thesis:_for_r_being_Real_ex_n_being_Element_of_NAT_st_ for_m_being_Element_of_NAT_st_n_<=_m_holds_ (f_/*_seq)_._m_<_r let r be Real; ::_thesis: ex n being Element of NAT st for m being Element of NAT st n <= m holds (f /* seq) . m < r consider g1 being set such that A6: g1 in (right_open_halfline r1) /\ (dom f) and A7: f . g1 < r by A2, RFUNCT_1:71; reconsider g1 = g1 as Real by A6; consider n being Element of NAT such that A8: for m being Element of NAT st n <= m holds (abs g1) + (abs r1) < seq . m by A4, Def4; take n = n; ::_thesis: for m being Element of NAT st n <= m holds (f /* seq) . m < r let m be Element of NAT ; ::_thesis: ( n <= m implies (f /* seq) . m < r ) assume n <= m ; ::_thesis: (f /* seq) . m < r then A9: (abs g1) + (abs r1) < seq . m by A8; ( r1 <= abs r1 & 0 <= abs g1 ) by ABSVALUE:4, COMPLEX1:46; then 0 + r1 <= (abs g1) + (abs r1) by XREAL_1:7; then r1 < seq . m by A9, XXREAL_0:2; then seq . m in { g2 where g2 is Real : r1 < g2 } ; then ( seq . m in rng seq & seq . m in right_open_halfline r1 ) by VALUED_0:28, XXREAL_1:230; then A10: seq . m in (right_open_halfline r1) /\ (dom f) by A5, XBOOLE_0:def_4; ( g1 <= abs g1 & 0 <= abs r1 ) by ABSVALUE:4, COMPLEX1:46; then g1 + 0 <= (abs g1) + (abs r1) by XREAL_1:7; then g1 < seq . m by A9, XXREAL_0:2; then f . (seq . m) <= f . g1 by A1, A6, A10, RFUNCT_2:23; then f . (seq . m) < r by A7, XXREAL_0:2; hence (f /* seq) . m < r by A5, FUNCT_2:108; ::_thesis: verum end; hence f /* seq is divergent_to-infty by Def5; ::_thesis: verum end; assume for r being Real ex g being Real st ( r < g & g in dom f ) ; ::_thesis: f is divergent_in+infty_to-infty hence f is divergent_in+infty_to-infty by A3, Def8; ::_thesis: verum end; theorem :: LIMFUNC1:65 for f being PartFunc of REAL,REAL st ex r being Real st ( f | (right_open_halfline r) is decreasing & not f | (right_open_halfline r) is bounded_below ) & ( for r being Real ex g being Real st ( r < g & g in dom f ) ) holds f is divergent_in+infty_to-infty by Th64; theorem Th66: :: LIMFUNC1:66 for f being PartFunc of REAL,REAL st ex r being Real st ( f | (left_open_halfline r) is non-increasing & not f | (left_open_halfline r) is bounded_above ) & ( for r being Real ex g being Real st ( g < r & g in dom f ) ) holds f is divergent_in-infty_to+infty proof let f be PartFunc of REAL,REAL; ::_thesis: ( ex r being Real st ( f | (left_open_halfline r) is non-increasing & not f | (left_open_halfline r) is bounded_above ) & ( for r being Real ex g being Real st ( g < r & g in dom f ) ) implies f is divergent_in-infty_to+infty ) given r1 being Real such that A1: f | (left_open_halfline r1) is non-increasing and A2: not f | (left_open_halfline r1) is bounded_above ; ::_thesis: ( ex r being Real st for g being Real holds ( not g < r or not g in dom f ) or f is divergent_in-infty_to+infty ) A3: now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_divergent_to-infty_&_rng_seq_c=_dom_f_holds_ f_/*_seq_is_divergent_to+infty let seq be Real_Sequence; ::_thesis: ( seq is divergent_to-infty & rng seq c= dom f implies f /* seq is divergent_to+infty ) assume that A4: seq is divergent_to-infty and A5: rng seq c= dom f ; ::_thesis: f /* seq is divergent_to+infty now__::_thesis:_for_r_being_Real_ex_n_being_Element_of_NAT_st_ for_m_being_Element_of_NAT_st_n_<=_m_holds_ r_<_(f_/*_seq)_._m let r be Real; ::_thesis: ex n being Element of NAT st for m being Element of NAT st n <= m holds r < (f /* seq) . m consider g1 being set such that A6: g1 in (left_open_halfline r1) /\ (dom f) and A7: r < f . g1 by A2, RFUNCT_1:70; reconsider g1 = g1 as Real by A6; consider n being Element of NAT such that A8: for m being Element of NAT st n <= m holds seq . m < (- (abs r1)) - (abs g1) by A4, Def5; take n = n; ::_thesis: for m being Element of NAT st n <= m holds r < (f /* seq) . m let m be Element of NAT ; ::_thesis: ( n <= m implies r < (f /* seq) . m ) assume n <= m ; ::_thesis: r < (f /* seq) . m then A9: seq . m < (- (abs r1)) - (abs g1) by A8; ( - (abs r1) <= r1 & 0 <= abs g1 ) by ABSVALUE:4, COMPLEX1:46; then (- (abs r1)) - (abs g1) <= r1 - 0 by XREAL_1:13; then seq . m < r1 by A9, XXREAL_0:2; then seq . m in { g2 where g2 is Real : g2 < r1 } ; then ( seq . m in rng seq & seq . m in left_open_halfline r1 ) by VALUED_0:28, XXREAL_1:229; then A10: seq . m in (left_open_halfline r1) /\ (dom f) by A5, XBOOLE_0:def_4; ( - (abs g1) <= g1 & 0 <= abs r1 ) by ABSVALUE:4, COMPLEX1:46; then (- (abs g1)) - (abs r1) <= g1 - 0 by XREAL_1:13; then seq . m < g1 by A9, XXREAL_0:2; then f . g1 <= f . (seq . m) by A1, A6, A10, RFUNCT_2:23; then r < f . (seq . m) by A7, XXREAL_0:2; hence r < (f /* seq) . m by A5, FUNCT_2:108; ::_thesis: verum end; hence f /* seq is divergent_to+infty by Def4; ::_thesis: verum end; assume for r being Real ex g being Real st ( g < r & g in dom f ) ; ::_thesis: f is divergent_in-infty_to+infty hence f is divergent_in-infty_to+infty by A3, Def10; ::_thesis: verum end; theorem :: LIMFUNC1:67 for f being PartFunc of REAL,REAL st ex r being Real st ( f | (left_open_halfline r) is decreasing & not f | (left_open_halfline r) is bounded_above ) & ( for r being Real ex g being Real st ( g < r & g in dom f ) ) holds f is divergent_in-infty_to+infty by Th66; theorem Th68: :: LIMFUNC1:68 for f being PartFunc of REAL,REAL st ex r being Real st ( f | (left_open_halfline r) is non-decreasing & not f | (left_open_halfline r) is bounded_below ) & ( for r being Real ex g being Real st ( g < r & g in dom f ) ) holds f is divergent_in-infty_to-infty proof let f be PartFunc of REAL,REAL; ::_thesis: ( ex r being Real st ( f | (left_open_halfline r) is non-decreasing & not f | (left_open_halfline r) is bounded_below ) & ( for r being Real ex g being Real st ( g < r & g in dom f ) ) implies f is divergent_in-infty_to-infty ) given r1 being Real such that A1: f | (left_open_halfline r1) is non-decreasing and A2: not f | (left_open_halfline r1) is bounded_below ; ::_thesis: ( ex r being Real st for g being Real holds ( not g < r or not g in dom f ) or f is divergent_in-infty_to-infty ) A3: now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_divergent_to-infty_&_rng_seq_c=_dom_f_holds_ f_/*_seq_is_divergent_to-infty let seq be Real_Sequence; ::_thesis: ( seq is divergent_to-infty & rng seq c= dom f implies f /* seq is divergent_to-infty ) assume that A4: seq is divergent_to-infty and A5: rng seq c= dom f ; ::_thesis: f /* seq is divergent_to-infty now__::_thesis:_for_r_being_Real_ex_n_being_Element_of_NAT_st_ for_m_being_Element_of_NAT_st_n_<=_m_holds_ (f_/*_seq)_._m_<_r let r be Real; ::_thesis: ex n being Element of NAT st for m being Element of NAT st n <= m holds (f /* seq) . m < r consider g1 being set such that A6: g1 in (left_open_halfline r1) /\ (dom f) and A7: f . g1 < r by A2, RFUNCT_1:71; reconsider g1 = g1 as Real by A6; consider n being Element of NAT such that A8: for m being Element of NAT st n <= m holds seq . m < (- (abs r1)) - (abs g1) by A4, Def5; take n = n; ::_thesis: for m being Element of NAT st n <= m holds (f /* seq) . m < r let m be Element of NAT ; ::_thesis: ( n <= m implies (f /* seq) . m < r ) assume n <= m ; ::_thesis: (f /* seq) . m < r then A9: seq . m < (- (abs r1)) - (abs g1) by A8; ( - (abs r1) <= r1 & 0 <= abs g1 ) by ABSVALUE:4, COMPLEX1:46; then (- (abs r1)) - (abs g1) <= r1 - 0 by XREAL_1:13; then seq . m < r1 by A9, XXREAL_0:2; then seq . m in { g2 where g2 is Real : g2 < r1 } ; then ( seq . m in rng seq & seq . m in left_open_halfline r1 ) by VALUED_0:28, XXREAL_1:229; then A10: seq . m in (left_open_halfline r1) /\ (dom f) by A5, XBOOLE_0:def_4; ( - (abs g1) <= g1 & 0 <= abs r1 ) by ABSVALUE:4, COMPLEX1:46; then (- (abs g1)) - (abs r1) <= g1 - 0 by XREAL_1:13; then seq . m < g1 by A9, XXREAL_0:2; then f . (seq . m) <= f . g1 by A1, A6, A10, RFUNCT_2:22; then f . (seq . m) < r by A7, XXREAL_0:2; hence (f /* seq) . m < r by A5, FUNCT_2:108; ::_thesis: verum end; hence f /* seq is divergent_to-infty by Def5; ::_thesis: verum end; assume for r being Real ex g being Real st ( g < r & g in dom f ) ; ::_thesis: f is divergent_in-infty_to-infty hence f is divergent_in-infty_to-infty by A3, Def11; ::_thesis: verum end; theorem :: LIMFUNC1:69 for f being PartFunc of REAL,REAL st ex r being Real st ( f | (left_open_halfline r) is increasing & not f | (left_open_halfline r) is bounded_below ) & ( for r being Real ex g being Real st ( g < r & g in dom f ) ) holds f is divergent_in-infty_to-infty by Th68; theorem Th70: :: LIMFUNC1:70 for f1, f being PartFunc of REAL,REAL st f1 is divergent_in+infty_to+infty & ( for r being Real ex g being Real st ( r < g & g in dom f ) ) & ex r being Real st ( (dom f) /\ (right_open_halfline r) c= (dom f1) /\ (right_open_halfline r) & ( for g being Real st g in (dom f) /\ (right_open_halfline r) holds f1 . g <= f . g ) ) holds f is divergent_in+infty_to+infty proof let f1, f be PartFunc of REAL,REAL; ::_thesis: ( f1 is divergent_in+infty_to+infty & ( for r being Real ex g being Real st ( r < g & g in dom f ) ) & ex r being Real st ( (dom f) /\ (right_open_halfline r) c= (dom f1) /\ (right_open_halfline r) & ( for g being Real st g in (dom f) /\ (right_open_halfline r) holds f1 . g <= f . g ) ) implies f is divergent_in+infty_to+infty ) assume that A1: f1 is divergent_in+infty_to+infty and A2: for r being Real ex g being Real st ( r < g & g in dom f ) ; ::_thesis: ( for r being Real holds ( not (dom f) /\ (right_open_halfline r) c= (dom f1) /\ (right_open_halfline r) or ex g being Real st ( g in (dom f) /\ (right_open_halfline r) & not f1 . g <= f . g ) ) or f is divergent_in+infty_to+infty ) given r1 being Real such that A3: (dom f) /\ (right_open_halfline r1) c= (dom f1) /\ (right_open_halfline r1) and A4: for g being Real st g in (dom f) /\ (right_open_halfline r1) holds f1 . g <= f . g ; ::_thesis: f is divergent_in+infty_to+infty now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_divergent_to+infty_&_rng_seq_c=_dom_f_holds_ f_/*_seq_is_divergent_to+infty let seq be Real_Sequence; ::_thesis: ( seq is divergent_to+infty & rng seq c= dom f implies f /* seq is divergent_to+infty ) assume that A5: seq is divergent_to+infty and A6: rng seq c= dom f ; ::_thesis: f /* seq is divergent_to+infty consider k being Element of NAT such that A7: for n being Element of NAT st k <= n holds r1 < seq . n by A5, Def4; now__::_thesis:_for_x_being_set_st_x_in_rng_(seq_^\_k)_holds_ x_in_right_open_halfline_r1 let x be set ; ::_thesis: ( x in rng (seq ^\ k) implies x in right_open_halfline r1 ) assume x in rng (seq ^\ k) ; ::_thesis: x in right_open_halfline r1 then consider n being Element of NAT such that A8: (seq ^\ k) . n = x by FUNCT_2:113; r1 < seq . (n + k) by A7, NAT_1:12; then r1 < (seq ^\ k) . n by NAT_1:def_3; then x in { g2 where g2 is Real : r1 < g2 } by A8; hence x in right_open_halfline r1 by XXREAL_1:230; ::_thesis: verum end; then A9: rng (seq ^\ k) c= right_open_halfline r1 by TARSKI:def_3; A10: rng (seq ^\ k) c= rng seq by VALUED_0:21; then rng (seq ^\ k) c= dom f by A6, XBOOLE_1:1; then A11: rng (seq ^\ k) c= (dom f) /\ (right_open_halfline r1) by A9, XBOOLE_1:19; then A12: rng (seq ^\ k) c= (dom f1) /\ (right_open_halfline r1) by A3, XBOOLE_1:1; A13: (dom f1) /\ (right_open_halfline r1) c= dom f1 by XBOOLE_1:17; A14: now__::_thesis:_for_n_being_Element_of_NAT_holds_(f1_/*_(seq_^\_k))_._n_<=_(f_/*_(seq_^\_k))_._n let n be Element of NAT ; ::_thesis: (f1 /* (seq ^\ k)) . n <= (f /* (seq ^\ k)) . n (seq ^\ k) . n in rng (seq ^\ k) by VALUED_0:28; then f1 . ((seq ^\ k) . n) <= f . ((seq ^\ k) . n) by A4, A11; then (f1 /* (seq ^\ k)) . n <= f . ((seq ^\ k) . n) by A12, A13, FUNCT_2:108, XBOOLE_1:1; hence (f1 /* (seq ^\ k)) . n <= (f /* (seq ^\ k)) . n by A6, A10, FUNCT_2:108, XBOOLE_1:1; ::_thesis: verum end; A15: seq ^\ k is divergent_to+infty by A5, Th26; rng (seq ^\ k) c= dom f1 by A12, A13, XBOOLE_1:1; then f1 /* (seq ^\ k) is divergent_to+infty by A1, A15, Def7; then A16: f /* (seq ^\ k) is divergent_to+infty by A14, Th42; f /* (seq ^\ k) = (f /* seq) ^\ k by A6, VALUED_0:27; hence f /* seq is divergent_to+infty by A16, Th7; ::_thesis: verum end; hence f is divergent_in+infty_to+infty by A2, Def7; ::_thesis: verum end; theorem Th71: :: LIMFUNC1:71 for f1, f being PartFunc of REAL,REAL st f1 is divergent_in+infty_to-infty & ( for r being Real ex g being Real st ( r < g & g in dom f ) ) & ex r being Real st ( (dom f) /\ (right_open_halfline r) c= (dom f1) /\ (right_open_halfline r) & ( for g being Real st g in (dom f) /\ (right_open_halfline r) holds f . g <= f1 . g ) ) holds f is divergent_in+infty_to-infty proof let f1, f be PartFunc of REAL,REAL; ::_thesis: ( f1 is divergent_in+infty_to-infty & ( for r being Real ex g being Real st ( r < g & g in dom f ) ) & ex r being Real st ( (dom f) /\ (right_open_halfline r) c= (dom f1) /\ (right_open_halfline r) & ( for g being Real st g in (dom f) /\ (right_open_halfline r) holds f . g <= f1 . g ) ) implies f is divergent_in+infty_to-infty ) assume that A1: f1 is divergent_in+infty_to-infty and A2: for r being Real ex g being Real st ( r < g & g in dom f ) ; ::_thesis: ( for r being Real holds ( not (dom f) /\ (right_open_halfline r) c= (dom f1) /\ (right_open_halfline r) or ex g being Real st ( g in (dom f) /\ (right_open_halfline r) & not f . g <= f1 . g ) ) or f is divergent_in+infty_to-infty ) given r1 being Real such that A3: (dom f) /\ (right_open_halfline r1) c= (dom f1) /\ (right_open_halfline r1) and A4: for g being Real st g in (dom f) /\ (right_open_halfline r1) holds f . g <= f1 . g ; ::_thesis: f is divergent_in+infty_to-infty now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_divergent_to+infty_&_rng_seq_c=_dom_f_holds_ f_/*_seq_is_divergent_to-infty let seq be Real_Sequence; ::_thesis: ( seq is divergent_to+infty & rng seq c= dom f implies f /* seq is divergent_to-infty ) assume that A5: seq is divergent_to+infty and A6: rng seq c= dom f ; ::_thesis: f /* seq is divergent_to-infty consider k being Element of NAT such that A7: for n being Element of NAT st k <= n holds r1 < seq . n by A5, Def4; now__::_thesis:_for_x_being_set_st_x_in_rng_(seq_^\_k)_holds_ x_in_right_open_halfline_r1 let x be set ; ::_thesis: ( x in rng (seq ^\ k) implies x in right_open_halfline r1 ) assume x in rng (seq ^\ k) ; ::_thesis: x in right_open_halfline r1 then consider n being Element of NAT such that A8: (seq ^\ k) . n = x by FUNCT_2:113; r1 < seq . (n + k) by A7, NAT_1:12; then r1 < (seq ^\ k) . n by NAT_1:def_3; then x in { g2 where g2 is Real : r1 < g2 } by A8; hence x in right_open_halfline r1 by XXREAL_1:230; ::_thesis: verum end; then A9: rng (seq ^\ k) c= right_open_halfline r1 by TARSKI:def_3; A10: rng (seq ^\ k) c= rng seq by VALUED_0:21; then rng (seq ^\ k) c= dom f by A6, XBOOLE_1:1; then A11: rng (seq ^\ k) c= (dom f) /\ (right_open_halfline r1) by A9, XBOOLE_1:19; then A12: rng (seq ^\ k) c= (dom f1) /\ (right_open_halfline r1) by A3, XBOOLE_1:1; A13: (dom f1) /\ (right_open_halfline r1) c= dom f1 by XBOOLE_1:17; A14: now__::_thesis:_for_n_being_Element_of_NAT_holds_(f_/*_(seq_^\_k))_._n_<=_(f1_/*_(seq_^\_k))_._n let n be Element of NAT ; ::_thesis: (f /* (seq ^\ k)) . n <= (f1 /* (seq ^\ k)) . n (seq ^\ k) . n in rng (seq ^\ k) by VALUED_0:28; then f . ((seq ^\ k) . n) <= f1 . ((seq ^\ k) . n) by A4, A11; then (f /* (seq ^\ k)) . n <= f1 . ((seq ^\ k) . n) by A6, A10, FUNCT_2:108, XBOOLE_1:1; hence (f /* (seq ^\ k)) . n <= (f1 /* (seq ^\ k)) . n by A12, A13, FUNCT_2:108, XBOOLE_1:1; ::_thesis: verum end; A15: seq ^\ k is divergent_to+infty by A5, Th26; rng (seq ^\ k) c= dom f1 by A12, A13, XBOOLE_1:1; then f1 /* (seq ^\ k) is divergent_to-infty by A1, A15, Def8; then A16: f /* (seq ^\ k) is divergent_to-infty by A14, Th43; f /* (seq ^\ k) = (f /* seq) ^\ k by A6, VALUED_0:27; hence f /* seq is divergent_to-infty by A16, Th7; ::_thesis: verum end; hence f is divergent_in+infty_to-infty by A2, Def8; ::_thesis: verum end; theorem Th72: :: LIMFUNC1:72 for f1, f being PartFunc of REAL,REAL st f1 is divergent_in-infty_to+infty & ( for r being Real ex g being Real st ( g < r & g in dom f ) ) & ex r being Real st ( (dom f) /\ (left_open_halfline r) c= (dom f1) /\ (left_open_halfline r) & ( for g being Real st g in (dom f) /\ (left_open_halfline r) holds f1 . g <= f . g ) ) holds f is divergent_in-infty_to+infty proof let f1, f be PartFunc of REAL,REAL; ::_thesis: ( f1 is divergent_in-infty_to+infty & ( for r being Real ex g being Real st ( g < r & g in dom f ) ) & ex r being Real st ( (dom f) /\ (left_open_halfline r) c= (dom f1) /\ (left_open_halfline r) & ( for g being Real st g in (dom f) /\ (left_open_halfline r) holds f1 . g <= f . g ) ) implies f is divergent_in-infty_to+infty ) assume that A1: f1 is divergent_in-infty_to+infty and A2: for r being Real ex g being Real st ( g < r & g in dom f ) ; ::_thesis: ( for r being Real holds ( not (dom f) /\ (left_open_halfline r) c= (dom f1) /\ (left_open_halfline r) or ex g being Real st ( g in (dom f) /\ (left_open_halfline r) & not f1 . g <= f . g ) ) or f is divergent_in-infty_to+infty ) given r1 being Real such that A3: (dom f) /\ (left_open_halfline r1) c= (dom f1) /\ (left_open_halfline r1) and A4: for g being Real st g in (dom f) /\ (left_open_halfline r1) holds f1 . g <= f . g ; ::_thesis: f is divergent_in-infty_to+infty now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_divergent_to-infty_&_rng_seq_c=_dom_f_holds_ f_/*_seq_is_divergent_to+infty let seq be Real_Sequence; ::_thesis: ( seq is divergent_to-infty & rng seq c= dom f implies f /* seq is divergent_to+infty ) assume that A5: seq is divergent_to-infty and A6: rng seq c= dom f ; ::_thesis: f /* seq is divergent_to+infty consider k being Element of NAT such that A7: for n being Element of NAT st k <= n holds seq . n < r1 by A5, Def5; now__::_thesis:_for_x_being_set_st_x_in_rng_(seq_^\_k)_holds_ x_in_left_open_halfline_r1 let x be set ; ::_thesis: ( x in rng (seq ^\ k) implies x in left_open_halfline r1 ) assume x in rng (seq ^\ k) ; ::_thesis: x in left_open_halfline r1 then consider n being Element of NAT such that A8: (seq ^\ k) . n = x by FUNCT_2:113; seq . (n + k) < r1 by A7, NAT_1:12; then (seq ^\ k) . n < r1 by NAT_1:def_3; then x in { g2 where g2 is Real : g2 < r1 } by A8; hence x in left_open_halfline r1 by XXREAL_1:229; ::_thesis: verum end; then A9: rng (seq ^\ k) c= left_open_halfline r1 by TARSKI:def_3; A10: rng (seq ^\ k) c= rng seq by VALUED_0:21; then rng (seq ^\ k) c= dom f by A6, XBOOLE_1:1; then A11: rng (seq ^\ k) c= (dom f) /\ (left_open_halfline r1) by A9, XBOOLE_1:19; then A12: rng (seq ^\ k) c= (dom f1) /\ (left_open_halfline r1) by A3, XBOOLE_1:1; A13: (dom f1) /\ (left_open_halfline r1) c= dom f1 by XBOOLE_1:17; A14: now__::_thesis:_for_n_being_Element_of_NAT_holds_(f1_/*_(seq_^\_k))_._n_<=_(f_/*_(seq_^\_k))_._n let n be Element of NAT ; ::_thesis: (f1 /* (seq ^\ k)) . n <= (f /* (seq ^\ k)) . n (seq ^\ k) . n in rng (seq ^\ k) by VALUED_0:28; then f1 . ((seq ^\ k) . n) <= f . ((seq ^\ k) . n) by A4, A11; then (f1 /* (seq ^\ k)) . n <= f . ((seq ^\ k) . n) by A12, A13, FUNCT_2:108, XBOOLE_1:1; hence (f1 /* (seq ^\ k)) . n <= (f /* (seq ^\ k)) . n by A6, A10, FUNCT_2:108, XBOOLE_1:1; ::_thesis: verum end; A15: seq ^\ k is divergent_to-infty by A5, Th27; rng (seq ^\ k) c= dom f1 by A12, A13, XBOOLE_1:1; then f1 /* (seq ^\ k) is divergent_to+infty by A1, A15, Def10; then A16: f /* (seq ^\ k) is divergent_to+infty by A14, Th42; f /* (seq ^\ k) = (f /* seq) ^\ k by A6, VALUED_0:27; hence f /* seq is divergent_to+infty by A16, Th7; ::_thesis: verum end; hence f is divergent_in-infty_to+infty by A2, Def10; ::_thesis: verum end; theorem Th73: :: LIMFUNC1:73 for f1, f being PartFunc of REAL,REAL st f1 is divergent_in-infty_to-infty & ( for r being Real ex g being Real st ( g < r & g in dom f ) ) & ex r being Real st ( (dom f) /\ (left_open_halfline r) c= (dom f1) /\ (left_open_halfline r) & ( for g being Real st g in (dom f) /\ (left_open_halfline r) holds f . g <= f1 . g ) ) holds f is divergent_in-infty_to-infty proof let f1, f be PartFunc of REAL,REAL; ::_thesis: ( f1 is divergent_in-infty_to-infty & ( for r being Real ex g being Real st ( g < r & g in dom f ) ) & ex r being Real st ( (dom f) /\ (left_open_halfline r) c= (dom f1) /\ (left_open_halfline r) & ( for g being Real st g in (dom f) /\ (left_open_halfline r) holds f . g <= f1 . g ) ) implies f is divergent_in-infty_to-infty ) assume that A1: f1 is divergent_in-infty_to-infty and A2: for r being Real ex g being Real st ( g < r & g in dom f ) ; ::_thesis: ( for r being Real holds ( not (dom f) /\ (left_open_halfline r) c= (dom f1) /\ (left_open_halfline r) or ex g being Real st ( g in (dom f) /\ (left_open_halfline r) & not f . g <= f1 . g ) ) or f is divergent_in-infty_to-infty ) given r1 being Real such that A3: (dom f) /\ (left_open_halfline r1) c= (dom f1) /\ (left_open_halfline r1) and A4: for g being Real st g in (dom f) /\ (left_open_halfline r1) holds f . g <= f1 . g ; ::_thesis: f is divergent_in-infty_to-infty now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_divergent_to-infty_&_rng_seq_c=_dom_f_holds_ f_/*_seq_is_divergent_to-infty let seq be Real_Sequence; ::_thesis: ( seq is divergent_to-infty & rng seq c= dom f implies f /* seq is divergent_to-infty ) assume that A5: seq is divergent_to-infty and A6: rng seq c= dom f ; ::_thesis: f /* seq is divergent_to-infty consider k being Element of NAT such that A7: for n being Element of NAT st k <= n holds seq . n < r1 by A5, Def5; now__::_thesis:_for_x_being_set_st_x_in_rng_(seq_^\_k)_holds_ x_in_left_open_halfline_r1 let x be set ; ::_thesis: ( x in rng (seq ^\ k) implies x in left_open_halfline r1 ) assume x in rng (seq ^\ k) ; ::_thesis: x in left_open_halfline r1 then consider n being Element of NAT such that A8: (seq ^\ k) . n = x by FUNCT_2:113; seq . (n + k) < r1 by A7, NAT_1:12; then (seq ^\ k) . n < r1 by NAT_1:def_3; then x in { g2 where g2 is Real : g2 < r1 } by A8; hence x in left_open_halfline r1 by XXREAL_1:229; ::_thesis: verum end; then A9: rng (seq ^\ k) c= left_open_halfline r1 by TARSKI:def_3; A10: rng (seq ^\ k) c= rng seq by VALUED_0:21; then rng (seq ^\ k) c= dom f by A6, XBOOLE_1:1; then A11: rng (seq ^\ k) c= (dom f) /\ (left_open_halfline r1) by A9, XBOOLE_1:19; then A12: rng (seq ^\ k) c= (dom f1) /\ (left_open_halfline r1) by A3, XBOOLE_1:1; A13: (dom f1) /\ (left_open_halfline r1) c= dom f1 by XBOOLE_1:17; A14: now__::_thesis:_for_n_being_Element_of_NAT_holds_(f_/*_(seq_^\_k))_._n_<=_(f1_/*_(seq_^\_k))_._n let n be Element of NAT ; ::_thesis: (f /* (seq ^\ k)) . n <= (f1 /* (seq ^\ k)) . n (seq ^\ k) . n in rng (seq ^\ k) by VALUED_0:28; then f . ((seq ^\ k) . n) <= f1 . ((seq ^\ k) . n) by A4, A11; then (f /* (seq ^\ k)) . n <= f1 . ((seq ^\ k) . n) by A6, A10, FUNCT_2:108, XBOOLE_1:1; hence (f /* (seq ^\ k)) . n <= (f1 /* (seq ^\ k)) . n by A12, A13, FUNCT_2:108, XBOOLE_1:1; ::_thesis: verum end; A15: seq ^\ k is divergent_to-infty by A5, Th27; rng (seq ^\ k) c= dom f1 by A12, A13, XBOOLE_1:1; then f1 /* (seq ^\ k) is divergent_to-infty by A1, A15, Def11; then A16: f /* (seq ^\ k) is divergent_to-infty by A14, Th43; f /* (seq ^\ k) = (f /* seq) ^\ k by A6, VALUED_0:27; hence f /* seq is divergent_to-infty by A16, Th7; ::_thesis: verum end; hence f is divergent_in-infty_to-infty by A2, Def11; ::_thesis: verum end; theorem :: LIMFUNC1:74 for f1, f being PartFunc of REAL,REAL st f1 is divergent_in+infty_to+infty & ex r being Real st ( right_open_halfline r c= (dom f) /\ (dom f1) & ( for g being Real st g in right_open_halfline r holds f1 . g <= f . g ) ) holds f is divergent_in+infty_to+infty proof let f1, f be PartFunc of REAL,REAL; ::_thesis: ( f1 is divergent_in+infty_to+infty & ex r being Real st ( right_open_halfline r c= (dom f) /\ (dom f1) & ( for g being Real st g in right_open_halfline r holds f1 . g <= f . g ) ) implies f is divergent_in+infty_to+infty ) assume A1: f1 is divergent_in+infty_to+infty ; ::_thesis: ( for r being Real holds ( not right_open_halfline r c= (dom f) /\ (dom f1) or ex g being Real st ( g in right_open_halfline r & not f1 . g <= f . g ) ) or f is divergent_in+infty_to+infty ) given r1 being Real such that A2: right_open_halfline r1 c= (dom f) /\ (dom f1) and A3: for g being Real st g in right_open_halfline r1 holds f1 . g <= f . g ; ::_thesis: f is divergent_in+infty_to+infty A4: now__::_thesis:_for_r_being_Real_ex_g_being_Real_st_ (_r_<_g_&_g_in_dom_f_) let r be Real; ::_thesis: ex g being Real st ( r < g & g in dom f ) consider g being real number such that A5: (abs r) + (abs r1) < g by XREAL_1:1; reconsider g = g as Real by XREAL_0:def_1; take g = g; ::_thesis: ( r < g & g in dom f ) ( 0 <= abs r1 & r <= abs r ) by ABSVALUE:4, COMPLEX1:46; then 0 + r <= (abs r) + (abs r1) by XREAL_1:7; hence r < g by A5, XXREAL_0:2; ::_thesis: g in dom f ( 0 <= abs r & r1 <= abs r1 ) by ABSVALUE:4, COMPLEX1:46; then 0 + r1 <= (abs r) + (abs r1) by XREAL_1:7; then r1 < g by A5, XXREAL_0:2; then g in { g2 where g2 is Real : r1 < g2 } ; then g in right_open_halfline r1 by XXREAL_1:230; hence g in dom f by A2, XBOOLE_0:def_4; ::_thesis: verum end; now__::_thesis:_(_(dom_f)_/\_(right_open_halfline_r1)_c=_(dom_f1)_/\_(right_open_halfline_r1)_&_(_for_g_being_Real_st_g_in_(dom_f)_/\_(right_open_halfline_r1)_holds_ f1_._g_<=_f_._g_)_) (dom f) /\ (dom f1) c= dom f by XBOOLE_1:17; then A6: (dom f) /\ (right_open_halfline r1) = right_open_halfline r1 by A2, XBOOLE_1:1, XBOOLE_1:28; (dom f) /\ (dom f1) c= dom f1 by XBOOLE_1:17; hence (dom f) /\ (right_open_halfline r1) c= (dom f1) /\ (right_open_halfline r1) by A2, A6, XBOOLE_1:1, XBOOLE_1:28; ::_thesis: for g being Real st g in (dom f) /\ (right_open_halfline r1) holds f1 . g <= f . g let g be Real; ::_thesis: ( g in (dom f) /\ (right_open_halfline r1) implies f1 . g <= f . g ) assume g in (dom f) /\ (right_open_halfline r1) ; ::_thesis: f1 . g <= f . g hence f1 . g <= f . g by A3, A6; ::_thesis: verum end; hence f is divergent_in+infty_to+infty by A1, A4, Th70; ::_thesis: verum end; theorem :: LIMFUNC1:75 for f1, f being PartFunc of REAL,REAL st f1 is divergent_in+infty_to-infty & ex r being Real st ( right_open_halfline r c= (dom f) /\ (dom f1) & ( for g being Real st g in right_open_halfline r holds f . g <= f1 . g ) ) holds f is divergent_in+infty_to-infty proof let f1, f be PartFunc of REAL,REAL; ::_thesis: ( f1 is divergent_in+infty_to-infty & ex r being Real st ( right_open_halfline r c= (dom f) /\ (dom f1) & ( for g being Real st g in right_open_halfline r holds f . g <= f1 . g ) ) implies f is divergent_in+infty_to-infty ) assume A1: f1 is divergent_in+infty_to-infty ; ::_thesis: ( for r being Real holds ( not right_open_halfline r c= (dom f) /\ (dom f1) or ex g being Real st ( g in right_open_halfline r & not f . g <= f1 . g ) ) or f is divergent_in+infty_to-infty ) given r1 being Real such that A2: right_open_halfline r1 c= (dom f) /\ (dom f1) and A3: for g being Real st g in right_open_halfline r1 holds f . g <= f1 . g ; ::_thesis: f is divergent_in+infty_to-infty A4: now__::_thesis:_for_r_being_Real_ex_g_being_Real_st_ (_r_<_g_&_g_in_dom_f_) let r be Real; ::_thesis: ex g being Real st ( r < g & g in dom f ) consider g being real number such that A5: (abs r) + (abs r1) < g by XREAL_1:1; reconsider g = g as Real by XREAL_0:def_1; take g = g; ::_thesis: ( r < g & g in dom f ) ( 0 <= abs r1 & r <= abs r ) by ABSVALUE:4, COMPLEX1:46; then 0 + r <= (abs r) + (abs r1) by XREAL_1:7; hence r < g by A5, XXREAL_0:2; ::_thesis: g in dom f ( 0 <= abs r & r1 <= abs r1 ) by ABSVALUE:4, COMPLEX1:46; then 0 + r1 <= (abs r) + (abs r1) by XREAL_1:7; then r1 < g by A5, XXREAL_0:2; then g in { g2 where g2 is Real : r1 < g2 } ; then g in right_open_halfline r1 by XXREAL_1:230; hence g in dom f by A2, XBOOLE_0:def_4; ::_thesis: verum end; now__::_thesis:_(_(dom_f)_/\_(right_open_halfline_r1)_c=_(dom_f1)_/\_(right_open_halfline_r1)_&_(_for_g_being_Real_st_g_in_(dom_f)_/\_(right_open_halfline_r1)_holds_ f_._g_<=_f1_._g_)_) (dom f) /\ (dom f1) c= dom f by XBOOLE_1:17; then A6: (dom f) /\ (right_open_halfline r1) = right_open_halfline r1 by A2, XBOOLE_1:1, XBOOLE_1:28; (dom f) /\ (dom f1) c= dom f1 by XBOOLE_1:17; hence (dom f) /\ (right_open_halfline r1) c= (dom f1) /\ (right_open_halfline r1) by A2, A6, XBOOLE_1:1, XBOOLE_1:28; ::_thesis: for g being Real st g in (dom f) /\ (right_open_halfline r1) holds f . g <= f1 . g let g be Real; ::_thesis: ( g in (dom f) /\ (right_open_halfline r1) implies f . g <= f1 . g ) assume g in (dom f) /\ (right_open_halfline r1) ; ::_thesis: f . g <= f1 . g hence f . g <= f1 . g by A3, A6; ::_thesis: verum end; hence f is divergent_in+infty_to-infty by A1, A4, Th71; ::_thesis: verum end; theorem :: LIMFUNC1:76 for f1, f being PartFunc of REAL,REAL st f1 is divergent_in-infty_to+infty & ex r being Real st ( left_open_halfline r c= (dom f) /\ (dom f1) & ( for g being Real st g in left_open_halfline r holds f1 . g <= f . g ) ) holds f is divergent_in-infty_to+infty proof let f1, f be PartFunc of REAL,REAL; ::_thesis: ( f1 is divergent_in-infty_to+infty & ex r being Real st ( left_open_halfline r c= (dom f) /\ (dom f1) & ( for g being Real st g in left_open_halfline r holds f1 . g <= f . g ) ) implies f is divergent_in-infty_to+infty ) assume A1: f1 is divergent_in-infty_to+infty ; ::_thesis: ( for r being Real holds ( not left_open_halfline r c= (dom f) /\ (dom f1) or ex g being Real st ( g in left_open_halfline r & not f1 . g <= f . g ) ) or f is divergent_in-infty_to+infty ) given r1 being Real such that A2: left_open_halfline r1 c= (dom f) /\ (dom f1) and A3: for g being Real st g in left_open_halfline r1 holds f1 . g <= f . g ; ::_thesis: f is divergent_in-infty_to+infty A4: now__::_thesis:_for_r_being_Real_ex_g_being_Real_st_ (_g_<_r_&_g_in_dom_f_) let r be Real; ::_thesis: ex g being Real st ( g < r & g in dom f ) consider g being real number such that A5: g < (- (abs r)) - (abs r1) by XREAL_1:2; reconsider g = g as Real by XREAL_0:def_1; take g = g; ::_thesis: ( g < r & g in dom f ) ( 0 <= abs r1 & - (abs r) <= r ) by ABSVALUE:4, COMPLEX1:46; then (- (abs r)) - (abs r1) <= r - 0 by XREAL_1:13; hence g < r by A5, XXREAL_0:2; ::_thesis: g in dom f ( 0 <= abs r & - (abs r1) <= r1 ) by ABSVALUE:4, COMPLEX1:46; then (- (abs r1)) - (abs r) <= r1 - 0 by XREAL_1:13; then g < r1 by A5, XXREAL_0:2; then g in { g2 where g2 is Real : g2 < r1 } ; then g in left_open_halfline r1 by XXREAL_1:229; hence g in dom f by A2, XBOOLE_0:def_4; ::_thesis: verum end; now__::_thesis:_(_(dom_f)_/\_(left_open_halfline_r1)_c=_(dom_f1)_/\_(left_open_halfline_r1)_&_(_for_g_being_Real_st_g_in_(dom_f)_/\_(left_open_halfline_r1)_holds_ f1_._g_<=_f_._g_)_) (dom f) /\ (dom f1) c= dom f by XBOOLE_1:17; then A6: (dom f) /\ (left_open_halfline r1) = left_open_halfline r1 by A2, XBOOLE_1:1, XBOOLE_1:28; (dom f) /\ (dom f1) c= dom f1 by XBOOLE_1:17; hence (dom f) /\ (left_open_halfline r1) c= (dom f1) /\ (left_open_halfline r1) by A2, A6, XBOOLE_1:1, XBOOLE_1:28; ::_thesis: for g being Real st g in (dom f) /\ (left_open_halfline r1) holds f1 . g <= f . g let g be Real; ::_thesis: ( g in (dom f) /\ (left_open_halfline r1) implies f1 . g <= f . g ) assume g in (dom f) /\ (left_open_halfline r1) ; ::_thesis: f1 . g <= f . g hence f1 . g <= f . g by A3, A6; ::_thesis: verum end; hence f is divergent_in-infty_to+infty by A1, A4, Th72; ::_thesis: verum end; theorem :: LIMFUNC1:77 for f1, f being PartFunc of REAL,REAL st f1 is divergent_in-infty_to-infty & ex r being Real st ( left_open_halfline r c= (dom f) /\ (dom f1) & ( for g being Real st g in left_open_halfline r holds f . g <= f1 . g ) ) holds f is divergent_in-infty_to-infty proof let f1, f be PartFunc of REAL,REAL; ::_thesis: ( f1 is divergent_in-infty_to-infty & ex r being Real st ( left_open_halfline r c= (dom f) /\ (dom f1) & ( for g being Real st g in left_open_halfline r holds f . g <= f1 . g ) ) implies f is divergent_in-infty_to-infty ) assume A1: f1 is divergent_in-infty_to-infty ; ::_thesis: ( for r being Real holds ( not left_open_halfline r c= (dom f) /\ (dom f1) or ex g being Real st ( g in left_open_halfline r & not f . g <= f1 . g ) ) or f is divergent_in-infty_to-infty ) given r1 being Real such that A2: left_open_halfline r1 c= (dom f) /\ (dom f1) and A3: for g being Real st g in left_open_halfline r1 holds f . g <= f1 . g ; ::_thesis: f is divergent_in-infty_to-infty A4: now__::_thesis:_for_r_being_Real_ex_g_being_Real_st_ (_g_<_r_&_g_in_dom_f_) let r be Real; ::_thesis: ex g being Real st ( g < r & g in dom f ) consider g being real number such that A5: g < (- (abs r)) - (abs r1) by XREAL_1:2; reconsider g = g as Real by XREAL_0:def_1; take g = g; ::_thesis: ( g < r & g in dom f ) ( 0 <= abs r1 & - (abs r) <= r ) by ABSVALUE:4, COMPLEX1:46; then (- (abs r)) - (abs r1) <= r - 0 by XREAL_1:13; hence g < r by A5, XXREAL_0:2; ::_thesis: g in dom f ( 0 <= abs r & - (abs r1) <= r1 ) by ABSVALUE:4, COMPLEX1:46; then (- (abs r1)) - (abs r) <= r1 - 0 by XREAL_1:13; then g < r1 by A5, XXREAL_0:2; then g in { g2 where g2 is Real : g2 < r1 } ; then g in left_open_halfline r1 by XXREAL_1:229; hence g in dom f by A2, XBOOLE_0:def_4; ::_thesis: verum end; now__::_thesis:_(_(dom_f)_/\_(left_open_halfline_r1)_c=_(dom_f1)_/\_(left_open_halfline_r1)_&_(_for_g_being_Real_st_g_in_(dom_f)_/\_(left_open_halfline_r1)_holds_ f_._g_<=_f1_._g_)_) (dom f) /\ (dom f1) c= dom f by XBOOLE_1:17; then A6: (dom f) /\ (left_open_halfline r1) = left_open_halfline r1 by A2, XBOOLE_1:1, XBOOLE_1:28; (dom f) /\ (dom f1) c= dom f1 by XBOOLE_1:17; hence (dom f) /\ (left_open_halfline r1) c= (dom f1) /\ (left_open_halfline r1) by A2, A6, XBOOLE_1:1, XBOOLE_1:28; ::_thesis: for g being Real st g in (dom f) /\ (left_open_halfline r1) holds f . g <= f1 . g let g be Real; ::_thesis: ( g in (dom f) /\ (left_open_halfline r1) implies f . g <= f1 . g ) assume g in (dom f) /\ (left_open_halfline r1) ; ::_thesis: f . g <= f1 . g hence f . g <= f1 . g by A3, A6; ::_thesis: verum end; hence f is divergent_in-infty_to-infty by A1, A4, Th73; ::_thesis: verum end; definition let f be PartFunc of REAL,REAL; assume A1: f is convergent_in+infty ; func lim_in+infty f -> Real means :Def12: :: LIMFUNC1:def 12 for seq being Real_Sequence st seq is divergent_to+infty & rng seq c= dom f holds ( f /* seq is convergent & lim (f /* seq) = it ); existence ex b1 being Real st for seq being Real_Sequence st seq is divergent_to+infty & rng seq c= dom f holds ( f /* seq is convergent & lim (f /* seq) = b1 ) by A1, Def6; uniqueness for b1, b2 being Real st ( for seq being Real_Sequence st seq is divergent_to+infty & rng seq c= dom f holds ( f /* seq is convergent & lim (f /* seq) = b1 ) ) & ( for seq being Real_Sequence st seq is divergent_to+infty & rng seq c= dom f holds ( f /* seq is convergent & lim (f /* seq) = b2 ) ) holds b1 = b2 proof defpred S1[ Element of NAT , real number ] means ( $1 < $2 & $2 in dom f ); let g1, g2 be Real; ::_thesis: ( ( for seq being Real_Sequence st seq is divergent_to+infty & rng seq c= dom f holds ( f /* seq is convergent & lim (f /* seq) = g1 ) ) & ( for seq being Real_Sequence st seq is divergent_to+infty & rng seq c= dom f holds ( f /* seq is convergent & lim (f /* seq) = g2 ) ) implies g1 = g2 ) assume that A2: for seq being Real_Sequence st seq is divergent_to+infty & rng seq c= dom f holds ( f /* seq is convergent & lim (f /* seq) = g1 ) and A3: for seq being Real_Sequence st seq is divergent_to+infty & rng seq c= dom f holds ( f /* seq is convergent & lim (f /* seq) = g2 ) ; ::_thesis: g1 = g2 A4: for n being Element of NAT ex r being Real st S1[n,r] by A1, Def6; consider s2 being Real_Sequence such that A5: for n being Element of NAT holds S1[n,s2 . n] from FUNCT_2:sch_3(A4); A6: rng s2 c= dom f proof let x be real number ; :: according to MEMBERED:def_9 ::_thesis: ( not x in rng s2 or x in dom f ) assume x in rng s2 ; ::_thesis: x in dom f then ex n being Element of NAT st x = s2 . n by FUNCT_2:113; hence x in dom f by A5; ::_thesis: verum end; now__::_thesis:_for_n_being_Element_of_NAT_holds_(incl_NAT)_._n_<=_s2_._n let n be Element of NAT ; ::_thesis: (incl NAT) . n <= s2 . n n < s2 . n by A5; hence (incl NAT) . n <= s2 . n by FUNCT_1:18; ::_thesis: verum end; then A7: s2 is divergent_to+infty by Lm4, Th20, Th42; then lim (f /* s2) = g1 by A2, A6; hence g1 = g2 by A3, A7, A6; ::_thesis: verum end; end; :: deftheorem Def12 defines lim_in+infty LIMFUNC1:def_12_:_ for f being PartFunc of REAL,REAL st f is convergent_in+infty holds for b2 being Real holds ( b2 = lim_in+infty f iff for seq being Real_Sequence st seq is divergent_to+infty & rng seq c= dom f holds ( f /* seq is convergent & lim (f /* seq) = b2 ) ); definition let f be PartFunc of REAL,REAL; assume A1: f is convergent_in-infty ; func lim_in-infty f -> Real means :Def13: :: LIMFUNC1:def 13 for seq being Real_Sequence st seq is divergent_to-infty & rng seq c= dom f holds ( f /* seq is convergent & lim (f /* seq) = it ); existence ex b1 being Real st for seq being Real_Sequence st seq is divergent_to-infty & rng seq c= dom f holds ( f /* seq is convergent & lim (f /* seq) = b1 ) by A1, Def9; uniqueness for b1, b2 being Real st ( for seq being Real_Sequence st seq is divergent_to-infty & rng seq c= dom f holds ( f /* seq is convergent & lim (f /* seq) = b1 ) ) & ( for seq being Real_Sequence st seq is divergent_to-infty & rng seq c= dom f holds ( f /* seq is convergent & lim (f /* seq) = b2 ) ) holds b1 = b2 proof deffunc H1( Element of NAT ) -> Element of REAL = - $1; defpred S1[ Element of NAT , real number ] means ( $2 < - $1 & $2 in dom f ); let g1, g2 be Real; ::_thesis: ( ( for seq being Real_Sequence st seq is divergent_to-infty & rng seq c= dom f holds ( f /* seq is convergent & lim (f /* seq) = g1 ) ) & ( for seq being Real_Sequence st seq is divergent_to-infty & rng seq c= dom f holds ( f /* seq is convergent & lim (f /* seq) = g2 ) ) implies g1 = g2 ) assume that A2: for seq being Real_Sequence st seq is divergent_to-infty & rng seq c= dom f holds ( f /* seq is convergent & lim (f /* seq) = g1 ) and A3: for seq being Real_Sequence st seq is divergent_to-infty & rng seq c= dom f holds ( f /* seq is convergent & lim (f /* seq) = g2 ) ; ::_thesis: g1 = g2 consider s2 being Real_Sequence such that A4: for n being Element of NAT holds s2 . n = H1(n) from SEQ_1:sch_1(); A5: for n being Element of NAT ex r being Real st S1[n,r] by A1, Def9; consider s1 being Real_Sequence such that A6: for n being Element of NAT holds S1[n,s1 . n] from FUNCT_2:sch_3(A5); A7: rng s1 c= dom f proof let x be real number ; :: according to MEMBERED:def_9 ::_thesis: ( not x in rng s1 or x in dom f ) assume x in rng s1 ; ::_thesis: x in dom f then ex n being Element of NAT st x = s1 . n by FUNCT_2:113; hence x in dom f by A6; ::_thesis: verum end; now__::_thesis:_for_n_being_Element_of_NAT_holds_s1_._n_<=_s2_._n let n be Element of NAT ; ::_thesis: s1 . n <= s2 . n s1 . n < - n by A6; hence s1 . n <= s2 . n by A4; ::_thesis: verum end; then A8: s1 is divergent_to-infty by A4, Th21, Th43; then lim (f /* s1) = g1 by A2, A7; hence g1 = g2 by A3, A8, A7; ::_thesis: verum end; end; :: deftheorem Def13 defines lim_in-infty LIMFUNC1:def_13_:_ for f being PartFunc of REAL,REAL st f is convergent_in-infty holds for b2 being Real holds ( b2 = lim_in-infty f iff for seq being Real_Sequence st seq is divergent_to-infty & rng seq c= dom f holds ( f /* seq is convergent & lim (f /* seq) = b2 ) ); theorem :: LIMFUNC1:78 for f being PartFunc of REAL,REAL for g being Real st f is convergent_in-infty holds ( lim_in-infty f = g iff for g1 being Real st 0 < g1 holds ex r being Real st for r1 being Real st r1 < r & r1 in dom f holds abs ((f . r1) - g) < g1 ) proof let f be PartFunc of REAL,REAL; ::_thesis: for g being Real st f is convergent_in-infty holds ( lim_in-infty f = g iff for g1 being Real st 0 < g1 holds ex r being Real st for r1 being Real st r1 < r & r1 in dom f holds abs ((f . r1) - g) < g1 ) let g be Real; ::_thesis: ( f is convergent_in-infty implies ( lim_in-infty f = g iff for g1 being Real st 0 < g1 holds ex r being Real st for r1 being Real st r1 < r & r1 in dom f holds abs ((f . r1) - g) < g1 ) ) assume A1: f is convergent_in-infty ; ::_thesis: ( lim_in-infty f = g iff for g1 being Real st 0 < g1 holds ex r being Real st for r1 being Real st r1 < r & r1 in dom f holds abs ((f . r1) - g) < g1 ) thus ( lim_in-infty f = g implies for g1 being Real st 0 < g1 holds ex r being Real st for r1 being Real st r1 < r & r1 in dom f holds abs ((f . r1) - g) < g1 ) ::_thesis: ( ( for g1 being Real st 0 < g1 holds ex r being Real st for r1 being Real st r1 < r & r1 in dom f holds abs ((f . r1) - g) < g1 ) implies lim_in-infty f = g ) proof deffunc H1( Element of NAT ) -> Element of REAL = - $1; assume A2: lim_in-infty f = g ; ::_thesis: for g1 being Real st 0 < g1 holds ex r being Real st for r1 being Real st r1 < r & r1 in dom f holds abs ((f . r1) - g) < g1 consider s1 being Real_Sequence such that A3: for n being Element of NAT holds s1 . n = H1(n) from SEQ_1:sch_1(); given g1 being Real such that A4: 0 < g1 and A5: for r being Real ex r1 being Real st ( r1 < r & r1 in dom f & abs ((f . r1) - g) >= g1 ) ; ::_thesis: contradiction defpred S1[ Element of NAT , real number ] means ( $2 < - $1 & $2 in dom f & abs ((f . $2) - g) >= g1 ); A6: for n being Element of NAT ex r being Real st S1[n,r] by A5; consider s being Real_Sequence such that A7: for n being Element of NAT holds S1[n,s . n] from FUNCT_2:sch_3(A6); now__::_thesis:_for_x_being_set_st_x_in_rng_s_holds_ x_in_dom_f let x be set ; ::_thesis: ( x in rng s implies x in dom f ) assume x in rng s ; ::_thesis: x in dom f then ex n being Element of NAT st s . n = x by FUNCT_2:113; hence x in dom f by A7; ::_thesis: verum end; then A8: rng s c= dom f by TARSKI:def_3; now__::_thesis:_for_n_being_Element_of_NAT_holds_s_._n_<=_s1_._n let n be Element of NAT ; ::_thesis: s . n <= s1 . n s . n < - n by A7; hence s . n <= s1 . n by A3; ::_thesis: verum end; then s is divergent_to-infty by A3, Th21, Th43; then ( f /* s is convergent & lim (f /* s) = g ) by A1, A2, A8, Def13; then consider n being Element of NAT such that A9: for m being Element of NAT st n <= m holds abs (((f /* s) . m) - g) < g1 by A4, SEQ_2:def_7; abs (((f /* s) . n) - g) < g1 by A9; then abs ((f . (s . n)) - g) < g1 by A8, FUNCT_2:108; hence contradiction by A7; ::_thesis: verum end; assume A10: for g1 being Real st 0 < g1 holds ex r being Real st for r1 being Real st r1 < r & r1 in dom f holds abs ((f . r1) - g) < g1 ; ::_thesis: lim_in-infty f = g now__::_thesis:_for_s_being_Real_Sequence_st_s_is_divergent_to-infty_&_rng_s_c=_dom_f_holds_ (_f_/*_s_is_convergent_&_lim_(f_/*_s)_=_g_) let s be Real_Sequence; ::_thesis: ( s is divergent_to-infty & rng s c= dom f implies ( f /* s is convergent & lim (f /* s) = g ) ) assume that A11: s is divergent_to-infty and A12: rng s c= dom f ; ::_thesis: ( f /* s is convergent & lim (f /* s) = g ) A13: now__::_thesis:_for_g1_being_real_number_st_0_<_g1_holds_ ex_n_being_Element_of_NAT_st_ for_m_being_Element_of_NAT_st_n_<=_m_holds_ abs_(((f_/*_s)_._m)_-_g)_<_g1 let g1 be real number ; ::_thesis: ( 0 < g1 implies ex n being Element of NAT st for m being Element of NAT st n <= m holds abs (((f /* s) . m) - g) < g1 ) assume A14: 0 < g1 ; ::_thesis: ex n being Element of NAT st for m being Element of NAT st n <= m holds abs (((f /* s) . m) - g) < g1 g1 is Real by XREAL_0:def_1; then consider r being Real such that A15: for r1 being Real st r1 < r & r1 in dom f holds abs ((f . r1) - g) < g1 by A10, A14; consider n being Element of NAT such that A16: for m being Element of NAT st n <= m holds s . m < r by A11, Def5; take n = n; ::_thesis: for m being Element of NAT st n <= m holds abs (((f /* s) . m) - g) < g1 let m be Element of NAT ; ::_thesis: ( n <= m implies abs (((f /* s) . m) - g) < g1 ) A17: s . m in rng s by VALUED_0:28; assume n <= m ; ::_thesis: abs (((f /* s) . m) - g) < g1 then abs ((f . (s . m)) - g) < g1 by A12, A15, A16, A17; hence abs (((f /* s) . m) - g) < g1 by A12, FUNCT_2:108; ::_thesis: verum end; hence f /* s is convergent by SEQ_2:def_6; ::_thesis: lim (f /* s) = g hence lim (f /* s) = g by A13, SEQ_2:def_7; ::_thesis: verum end; hence lim_in-infty f = g by A1, Def13; ::_thesis: verum end; theorem :: LIMFUNC1:79 for f being PartFunc of REAL,REAL for g being Real st f is convergent_in+infty holds ( lim_in+infty f = g iff for g1 being Real st 0 < g1 holds ex r being Real st for r1 being Real st r < r1 & r1 in dom f holds abs ((f . r1) - g) < g1 ) proof let f be PartFunc of REAL,REAL; ::_thesis: for g being Real st f is convergent_in+infty holds ( lim_in+infty f = g iff for g1 being Real st 0 < g1 holds ex r being Real st for r1 being Real st r < r1 & r1 in dom f holds abs ((f . r1) - g) < g1 ) let g be Real; ::_thesis: ( f is convergent_in+infty implies ( lim_in+infty f = g iff for g1 being Real st 0 < g1 holds ex r being Real st for r1 being Real st r < r1 & r1 in dom f holds abs ((f . r1) - g) < g1 ) ) assume A1: f is convergent_in+infty ; ::_thesis: ( lim_in+infty f = g iff for g1 being Real st 0 < g1 holds ex r being Real st for r1 being Real st r < r1 & r1 in dom f holds abs ((f . r1) - g) < g1 ) thus ( lim_in+infty f = g implies for g1 being Real st 0 < g1 holds ex r being Real st for r1 being Real st r < r1 & r1 in dom f holds abs ((f . r1) - g) < g1 ) ::_thesis: ( ( for g1 being Real st 0 < g1 holds ex r being Real st for r1 being Real st r < r1 & r1 in dom f holds abs ((f . r1) - g) < g1 ) implies lim_in+infty f = g ) proof assume A2: lim_in+infty f = g ; ::_thesis: for g1 being Real st 0 < g1 holds ex r being Real st for r1 being Real st r < r1 & r1 in dom f holds abs ((f . r1) - g) < g1 given g1 being Real such that A3: 0 < g1 and A4: for r being Real ex r1 being Real st ( r < r1 & r1 in dom f & abs ((f . r1) - g) >= g1 ) ; ::_thesis: contradiction defpred S1[ Element of NAT , real number ] means ( $1 < $2 & $2 in dom f & abs ((f . $2) - g) >= g1 ); A5: for n being Element of NAT ex r being Real st S1[n,r] by A4; consider s being Real_Sequence such that A6: for n being Element of NAT holds S1[n,s . n] from FUNCT_2:sch_3(A5); now__::_thesis:_for_x_being_set_st_x_in_rng_s_holds_ x_in_dom_f let x be set ; ::_thesis: ( x in rng s implies x in dom f ) assume x in rng s ; ::_thesis: x in dom f then ex n being Element of NAT st s . n = x by FUNCT_2:113; hence x in dom f by A6; ::_thesis: verum end; then A7: rng s c= dom f by TARSKI:def_3; now__::_thesis:_for_n_being_Element_of_NAT_holds_(incl_NAT)_._n_<=_s_._n let n be Element of NAT ; ::_thesis: (incl NAT) . n <= s . n n < s . n by A6; hence (incl NAT) . n <= s . n by FUNCT_1:18; ::_thesis: verum end; then s is divergent_to+infty by Lm4, Th20, Th42; then ( f /* s is convergent & lim (f /* s) = g ) by A1, A2, A7, Def12; then consider n being Element of NAT such that A8: for m being Element of NAT st n <= m holds abs (((f /* s) . m) - g) < g1 by A3, SEQ_2:def_7; abs (((f /* s) . n) - g) < g1 by A8; then abs ((f . (s . n)) - g) < g1 by A7, FUNCT_2:108; hence contradiction by A6; ::_thesis: verum end; assume A9: for g1 being Real st 0 < g1 holds ex r being Real st for r1 being Real st r < r1 & r1 in dom f holds abs ((f . r1) - g) < g1 ; ::_thesis: lim_in+infty f = g now__::_thesis:_for_s_being_Real_Sequence_st_s_is_divergent_to+infty_&_rng_s_c=_dom_f_holds_ (_f_/*_s_is_convergent_&_lim_(f_/*_s)_=_g_) let s be Real_Sequence; ::_thesis: ( s is divergent_to+infty & rng s c= dom f implies ( f /* s is convergent & lim (f /* s) = g ) ) assume that A10: s is divergent_to+infty and A11: rng s c= dom f ; ::_thesis: ( f /* s is convergent & lim (f /* s) = g ) A12: now__::_thesis:_for_g1_being_real_number_st_0_<_g1_holds_ ex_n_being_Element_of_NAT_st_ for_m_being_Element_of_NAT_st_n_<=_m_holds_ abs_(((f_/*_s)_._m)_-_g)_<_g1 let g1 be real number ; ::_thesis: ( 0 < g1 implies ex n being Element of NAT st for m being Element of NAT st n <= m holds abs (((f /* s) . m) - g) < g1 ) assume A13: 0 < g1 ; ::_thesis: ex n being Element of NAT st for m being Element of NAT st n <= m holds abs (((f /* s) . m) - g) < g1 g1 is Real by XREAL_0:def_1; then consider r being Real such that A14: for r1 being Real st r < r1 & r1 in dom f holds abs ((f . r1) - g) < g1 by A9, A13; consider n being Element of NAT such that A15: for m being Element of NAT st n <= m holds r < s . m by A10, Def4; take n = n; ::_thesis: for m being Element of NAT st n <= m holds abs (((f /* s) . m) - g) < g1 let m be Element of NAT ; ::_thesis: ( n <= m implies abs (((f /* s) . m) - g) < g1 ) A16: s . m in rng s by VALUED_0:28; assume n <= m ; ::_thesis: abs (((f /* s) . m) - g) < g1 then abs ((f . (s . m)) - g) < g1 by A11, A14, A15, A16; hence abs (((f /* s) . m) - g) < g1 by A11, FUNCT_2:108; ::_thesis: verum end; hence f /* s is convergent by SEQ_2:def_6; ::_thesis: lim (f /* s) = g hence lim (f /* s) = g by A12, SEQ_2:def_7; ::_thesis: verum end; hence lim_in+infty f = g by A1, Def12; ::_thesis: verum end; theorem Th80: :: LIMFUNC1:80 for f being PartFunc of REAL,REAL for r being Real st f is convergent_in+infty holds ( r (#) f is convergent_in+infty & lim_in+infty (r (#) f) = r * (lim_in+infty f) ) proof let f be PartFunc of REAL,REAL; ::_thesis: for r being Real st f is convergent_in+infty holds ( r (#) f is convergent_in+infty & lim_in+infty (r (#) f) = r * (lim_in+infty f) ) let r be Real; ::_thesis: ( f is convergent_in+infty implies ( r (#) f is convergent_in+infty & lim_in+infty (r (#) f) = r * (lim_in+infty f) ) ) assume A1: f is convergent_in+infty ; ::_thesis: ( r (#) f is convergent_in+infty & lim_in+infty (r (#) f) = r * (lim_in+infty f) ) A2: now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_divergent_to+infty_&_rng_seq_c=_dom_(r_(#)_f)_holds_ (_(r_(#)_f)_/*_seq_is_convergent_&_lim_((r_(#)_f)_/*_seq)_=_r_*_(lim_in+infty_f)_) let seq be Real_Sequence; ::_thesis: ( seq is divergent_to+infty & rng seq c= dom (r (#) f) implies ( (r (#) f) /* seq is convergent & lim ((r (#) f) /* seq) = r * (lim_in+infty f) ) ) assume that A3: seq is divergent_to+infty and A4: rng seq c= dom (r (#) f) ; ::_thesis: ( (r (#) f) /* seq is convergent & lim ((r (#) f) /* seq) = r * (lim_in+infty f) ) A5: rng seq c= dom f by A4, VALUED_1:def_5; then A6: lim (f /* seq) = lim_in+infty f by A1, A3, Def12; lim_in+infty f = lim_in+infty f ; then A7: f /* seq is convergent by A1, A3, A5, Def12; then r (#) (f /* seq) is convergent ; hence (r (#) f) /* seq is convergent by A5, RFUNCT_2:9; ::_thesis: lim ((r (#) f) /* seq) = r * (lim_in+infty f) thus lim ((r (#) f) /* seq) = lim (r (#) (f /* seq)) by A5, RFUNCT_2:9 .= r * (lim_in+infty f) by A7, A6, SEQ_2:8 ; ::_thesis: verum end; for r1 being Real ex g being Real st ( r1 < g & g in dom (r (#) f) ) proof let r1 be Real; ::_thesis: ex g being Real st ( r1 < g & g in dom (r (#) f) ) consider g being Real such that A8: ( r1 < g & g in dom f ) by A1, Def6; take g ; ::_thesis: ( r1 < g & g in dom (r (#) f) ) thus ( r1 < g & g in dom (r (#) f) ) by A8, VALUED_1:def_5; ::_thesis: verum end; hence r (#) f is convergent_in+infty by A2, Def6; ::_thesis: lim_in+infty (r (#) f) = r * (lim_in+infty f) hence lim_in+infty (r (#) f) = r * (lim_in+infty f) by A2, Def12; ::_thesis: verum end; theorem Th81: :: LIMFUNC1:81 for f being PartFunc of REAL,REAL st f is convergent_in+infty holds ( - f is convergent_in+infty & lim_in+infty (- f) = - (lim_in+infty f) ) proof let f be PartFunc of REAL,REAL; ::_thesis: ( f is convergent_in+infty implies ( - f is convergent_in+infty & lim_in+infty (- f) = - (lim_in+infty f) ) ) assume A1: f is convergent_in+infty ; ::_thesis: ( - f is convergent_in+infty & lim_in+infty (- f) = - (lim_in+infty f) ) (- 1) (#) f = - f ; hence - f is convergent_in+infty by A1, Th80; ::_thesis: lim_in+infty (- f) = - (lim_in+infty f) thus lim_in+infty (- f) = (- 1) * (lim_in+infty f) by A1, Th80 .= - (lim_in+infty f) ; ::_thesis: verum end; theorem Th82: :: LIMFUNC1:82 for f1, f2 being PartFunc of REAL,REAL st f1 is convergent_in+infty & f2 is convergent_in+infty & ( for r being Real ex g being Real st ( r < g & g in dom (f1 + f2) ) ) holds ( f1 + f2 is convergent_in+infty & lim_in+infty (f1 + f2) = (lim_in+infty f1) + (lim_in+infty f2) ) proof let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is convergent_in+infty & f2 is convergent_in+infty & ( for r being Real ex g being Real st ( r < g & g in dom (f1 + f2) ) ) implies ( f1 + f2 is convergent_in+infty & lim_in+infty (f1 + f2) = (lim_in+infty f1) + (lim_in+infty f2) ) ) assume that A1: f1 is convergent_in+infty and A2: f2 is convergent_in+infty and A3: for r being Real ex g being Real st ( r < g & g in dom (f1 + f2) ) ; ::_thesis: ( f1 + f2 is convergent_in+infty & lim_in+infty (f1 + f2) = (lim_in+infty f1) + (lim_in+infty f2) ) A4: now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_divergent_to+infty_&_rng_seq_c=_dom_(f1_+_f2)_holds_ (_(f1_+_f2)_/*_seq_is_convergent_&_lim_((f1_+_f2)_/*_seq)_=_(lim_in+infty_f1)_+_(lim_in+infty_f2)_) let seq be Real_Sequence; ::_thesis: ( seq is divergent_to+infty & rng seq c= dom (f1 + f2) implies ( (f1 + f2) /* seq is convergent & lim ((f1 + f2) /* seq) = (lim_in+infty f1) + (lim_in+infty f2) ) ) assume that A5: seq is divergent_to+infty and A6: rng seq c= dom (f1 + f2) ; ::_thesis: ( (f1 + f2) /* seq is convergent & lim ((f1 + f2) /* seq) = (lim_in+infty f1) + (lim_in+infty f2) ) A7: dom (f1 + f2) = (dom f1) /\ (dom f2) by A6, Lm2; A8: rng seq c= dom f2 by A6, Lm2; then A9: lim (f2 /* seq) = lim_in+infty f2 by A2, A5, Def12; lim_in+infty f2 = lim_in+infty f2 ; then A10: f2 /* seq is convergent by A2, A5, A8, Def12; A11: rng seq c= dom f1 by A6, Lm2; then A12: lim (f1 /* seq) = lim_in+infty f1 by A1, A5, Def12; lim_in+infty f1 = lim_in+infty f1 ; then A13: f1 /* seq is convergent by A1, A5, A11, Def12; then (f1 /* seq) + (f2 /* seq) is convergent by A10; hence (f1 + f2) /* seq is convergent by A6, A7, RFUNCT_2:8; ::_thesis: lim ((f1 + f2) /* seq) = (lim_in+infty f1) + (lim_in+infty f2) thus lim ((f1 + f2) /* seq) = lim ((f1 /* seq) + (f2 /* seq)) by A6, A7, RFUNCT_2:8 .= (lim_in+infty f1) + (lim_in+infty f2) by A13, A12, A10, A9, SEQ_2:6 ; ::_thesis: verum end; hence f1 + f2 is convergent_in+infty by A3, Def6; ::_thesis: lim_in+infty (f1 + f2) = (lim_in+infty f1) + (lim_in+infty f2) hence lim_in+infty (f1 + f2) = (lim_in+infty f1) + (lim_in+infty f2) by A4, Def12; ::_thesis: verum end; theorem :: LIMFUNC1:83 for f1, f2 being PartFunc of REAL,REAL st f1 is convergent_in+infty & f2 is convergent_in+infty & ( for r being Real ex g being Real st ( r < g & g in dom (f1 - f2) ) ) holds ( f1 - f2 is convergent_in+infty & lim_in+infty (f1 - f2) = (lim_in+infty f1) - (lim_in+infty f2) ) proof let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is convergent_in+infty & f2 is convergent_in+infty & ( for r being Real ex g being Real st ( r < g & g in dom (f1 - f2) ) ) implies ( f1 - f2 is convergent_in+infty & lim_in+infty (f1 - f2) = (lim_in+infty f1) - (lim_in+infty f2) ) ) assume that A1: f1 is convergent_in+infty and A2: f2 is convergent_in+infty and A3: for r being Real ex g being Real st ( r < g & g in dom (f1 - f2) ) ; ::_thesis: ( f1 - f2 is convergent_in+infty & lim_in+infty (f1 - f2) = (lim_in+infty f1) - (lim_in+infty f2) ) A4: - f2 is convergent_in+infty by A2, Th81; hence f1 - f2 is convergent_in+infty by A1, A3, Th82; ::_thesis: lim_in+infty (f1 - f2) = (lim_in+infty f1) - (lim_in+infty f2) lim_in+infty (- f2) = - (lim_in+infty f2) by A2, Th81; hence lim_in+infty (f1 - f2) = (lim_in+infty f1) + (- (lim_in+infty f2)) by A1, A3, A4, Th82 .= (lim_in+infty f1) - (lim_in+infty f2) ; ::_thesis: verum end; theorem :: LIMFUNC1:84 for f being PartFunc of REAL,REAL st f is convergent_in+infty & f " {0} = {} & lim_in+infty f <> 0 holds ( f ^ is convergent_in+infty & lim_in+infty (f ^) = (lim_in+infty f) " ) proof let f be PartFunc of REAL,REAL; ::_thesis: ( f is convergent_in+infty & f " {0} = {} & lim_in+infty f <> 0 implies ( f ^ is convergent_in+infty & lim_in+infty (f ^) = (lim_in+infty f) " ) ) assume that A1: f is convergent_in+infty and A2: f " {0} = {} and A3: lim_in+infty f <> 0 ; ::_thesis: ( f ^ is convergent_in+infty & lim_in+infty (f ^) = (lim_in+infty f) " ) A4: dom f = (dom f) \ (f " {0}) by A2 .= dom (f ^) by RFUNCT_1:def_2 ; A5: now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_divergent_to+infty_&_rng_seq_c=_dom_(f_^)_holds_ (_(f_^)_/*_seq_is_convergent_&_lim_((f_^)_/*_seq)_=_(lim_in+infty_f)_"_) let seq be Real_Sequence; ::_thesis: ( seq is divergent_to+infty & rng seq c= dom (f ^) implies ( (f ^) /* seq is convergent & lim ((f ^) /* seq) = (lim_in+infty f) " ) ) assume that A6: seq is divergent_to+infty and A7: rng seq c= dom (f ^) ; ::_thesis: ( (f ^) /* seq is convergent & lim ((f ^) /* seq) = (lim_in+infty f) " ) A8: ( f /* seq is convergent & lim (f /* seq) = lim_in+infty f ) by A1, A3, A4, A6, A7, Def12; then (f /* seq) " is convergent by A3, A7, RFUNCT_2:11, SEQ_2:21; hence (f ^) /* seq is convergent by A7, RFUNCT_2:12; ::_thesis: lim ((f ^) /* seq) = (lim_in+infty f) " thus lim ((f ^) /* seq) = lim ((f /* seq) ") by A7, RFUNCT_2:12 .= (lim_in+infty f) " by A3, A7, A8, RFUNCT_2:11, SEQ_2:22 ; ::_thesis: verum end; for r being Real ex g being Real st ( r < g & g in dom (f ^) ) by A1, A4, Def6; hence f ^ is convergent_in+infty by A5, Def6; ::_thesis: lim_in+infty (f ^) = (lim_in+infty f) " hence lim_in+infty (f ^) = (lim_in+infty f) " by A5, Def12; ::_thesis: verum end; theorem :: LIMFUNC1:85 for f being PartFunc of REAL,REAL st f is convergent_in+infty holds ( abs f is convergent_in+infty & lim_in+infty (abs f) = abs (lim_in+infty f) ) proof let f be PartFunc of REAL,REAL; ::_thesis: ( f is convergent_in+infty implies ( abs f is convergent_in+infty & lim_in+infty (abs f) = abs (lim_in+infty f) ) ) assume A1: f is convergent_in+infty ; ::_thesis: ( abs f is convergent_in+infty & lim_in+infty (abs f) = abs (lim_in+infty f) ) A2: now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_divergent_to+infty_&_rng_seq_c=_dom_(abs_f)_holds_ (_(abs_f)_/*_seq_is_convergent_&_lim_((abs_f)_/*_seq)_=_abs_(lim_in+infty_f)_) let seq be Real_Sequence; ::_thesis: ( seq is divergent_to+infty & rng seq c= dom (abs f) implies ( (abs f) /* seq is convergent & lim ((abs f) /* seq) = abs (lim_in+infty f) ) ) assume that A3: seq is divergent_to+infty and A4: rng seq c= dom (abs f) ; ::_thesis: ( (abs f) /* seq is convergent & lim ((abs f) /* seq) = abs (lim_in+infty f) ) A5: rng seq c= dom f by A4, VALUED_1:def_11; then A6: lim (f /* seq) = lim_in+infty f by A1, A3, Def12; lim_in+infty f = lim_in+infty f ; then A7: f /* seq is convergent by A1, A3, A5, Def12; then abs (f /* seq) is convergent ; hence (abs f) /* seq is convergent by A5, RFUNCT_2:10; ::_thesis: lim ((abs f) /* seq) = abs (lim_in+infty f) thus lim ((abs f) /* seq) = lim (abs (f /* seq)) by A5, RFUNCT_2:10 .= abs (lim_in+infty f) by A7, A6, SEQ_4:14 ; ::_thesis: verum end; now__::_thesis:_for_r_being_Real_ex_g_being_Real_st_ (_r_<_g_&_g_in_dom_(abs_f)_) let r be Real; ::_thesis: ex g being Real st ( r < g & g in dom (abs f) ) consider g being Real such that A8: ( r < g & g in dom f ) by A1, Def6; take g = g; ::_thesis: ( r < g & g in dom (abs f) ) thus ( r < g & g in dom (abs f) ) by A8, VALUED_1:def_11; ::_thesis: verum end; hence abs f is convergent_in+infty by A2, Def6; ::_thesis: lim_in+infty (abs f) = abs (lim_in+infty f) hence lim_in+infty (abs f) = abs (lim_in+infty f) by A2, Def12; ::_thesis: verum end; theorem Th86: :: LIMFUNC1:86 for f being PartFunc of REAL,REAL st f is convergent_in+infty & lim_in+infty f <> 0 & ( for r being Real ex g being Real st ( r < g & g in dom f & f . g <> 0 ) ) holds ( f ^ is convergent_in+infty & lim_in+infty (f ^) = (lim_in+infty f) " ) proof let f be PartFunc of REAL,REAL; ::_thesis: ( f is convergent_in+infty & lim_in+infty f <> 0 & ( for r being Real ex g being Real st ( r < g & g in dom f & f . g <> 0 ) ) implies ( f ^ is convergent_in+infty & lim_in+infty (f ^) = (lim_in+infty f) " ) ) assume that A1: f is convergent_in+infty and A2: lim_in+infty f <> 0 and A3: for r being Real ex g being Real st ( r < g & g in dom f & f . g <> 0 ) ; ::_thesis: ( f ^ is convergent_in+infty & lim_in+infty (f ^) = (lim_in+infty f) " ) A4: now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_divergent_to+infty_&_rng_seq_c=_dom_(f_^)_holds_ (_(f_^)_/*_seq_is_convergent_&_lim_((f_^)_/*_seq)_=_(lim_in+infty_f)_"_) let seq be Real_Sequence; ::_thesis: ( seq is divergent_to+infty & rng seq c= dom (f ^) implies ( (f ^) /* seq is convergent & lim ((f ^) /* seq) = (lim_in+infty f) " ) ) assume that A5: seq is divergent_to+infty and A6: rng seq c= dom (f ^) ; ::_thesis: ( (f ^) /* seq is convergent & lim ((f ^) /* seq) = (lim_in+infty f) " ) ( dom (f ^) = (dom f) \ (f " {0}) & (dom f) \ (f " {0}) c= dom f ) by RFUNCT_1:def_2, XBOOLE_1:36; then rng seq c= dom f by A6, XBOOLE_1:1; then A7: ( f /* seq is convergent & lim (f /* seq) = lim_in+infty f ) by A1, A2, A5, Def12; then (f /* seq) " is convergent by A2, A6, RFUNCT_2:11, SEQ_2:21; hence (f ^) /* seq is convergent by A6, RFUNCT_2:12; ::_thesis: lim ((f ^) /* seq) = (lim_in+infty f) " thus lim ((f ^) /* seq) = lim ((f /* seq) ") by A6, RFUNCT_2:12 .= (lim_in+infty f) " by A2, A6, A7, RFUNCT_2:11, SEQ_2:22 ; ::_thesis: verum end; now__::_thesis:_for_r_being_Real_ex_g_being_Real_st_ (_r_<_g_&_g_in_dom_(f_^)_) let r be Real; ::_thesis: ex g being Real st ( r < g & g in dom (f ^) ) consider g being Real such that A8: r < g and A9: g in dom f and A10: f . g <> 0 by A3; take g = g; ::_thesis: ( r < g & g in dom (f ^) ) not f . g in {0} by A10, TARSKI:def_1; then not g in f " {0} by FUNCT_1:def_7; then g in (dom f) \ (f " {0}) by A9, XBOOLE_0:def_5; hence ( r < g & g in dom (f ^) ) by A8, RFUNCT_1:def_2; ::_thesis: verum end; hence f ^ is convergent_in+infty by A4, Def6; ::_thesis: lim_in+infty (f ^) = (lim_in+infty f) " hence lim_in+infty (f ^) = (lim_in+infty f) " by A4, Def12; ::_thesis: verum end; theorem Th87: :: LIMFUNC1:87 for f1, f2 being PartFunc of REAL,REAL st f1 is convergent_in+infty & f2 is convergent_in+infty & ( for r being Real ex g being Real st ( r < g & g in dom (f1 (#) f2) ) ) holds ( f1 (#) f2 is convergent_in+infty & lim_in+infty (f1 (#) f2) = (lim_in+infty f1) * (lim_in+infty f2) ) proof let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is convergent_in+infty & f2 is convergent_in+infty & ( for r being Real ex g being Real st ( r < g & g in dom (f1 (#) f2) ) ) implies ( f1 (#) f2 is convergent_in+infty & lim_in+infty (f1 (#) f2) = (lim_in+infty f1) * (lim_in+infty f2) ) ) assume that A1: f1 is convergent_in+infty and A2: f2 is convergent_in+infty and A3: for r being Real ex g being Real st ( r < g & g in dom (f1 (#) f2) ) ; ::_thesis: ( f1 (#) f2 is convergent_in+infty & lim_in+infty (f1 (#) f2) = (lim_in+infty f1) * (lim_in+infty f2) ) A4: now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_divergent_to+infty_&_rng_seq_c=_dom_(f1_(#)_f2)_holds_ (_(f1_(#)_f2)_/*_seq_is_convergent_&_lim_((f1_(#)_f2)_/*_seq)_=_(lim_in+infty_f1)_*_(lim_in+infty_f2)_) let seq be Real_Sequence; ::_thesis: ( seq is divergent_to+infty & rng seq c= dom (f1 (#) f2) implies ( (f1 (#) f2) /* seq is convergent & lim ((f1 (#) f2) /* seq) = (lim_in+infty f1) * (lim_in+infty f2) ) ) assume that A5: seq is divergent_to+infty and A6: rng seq c= dom (f1 (#) f2) ; ::_thesis: ( (f1 (#) f2) /* seq is convergent & lim ((f1 (#) f2) /* seq) = (lim_in+infty f1) * (lim_in+infty f2) ) A7: dom (f1 (#) f2) = (dom f1) /\ (dom f2) by A6, Lm3; A8: rng seq c= dom f2 by A6, Lm3; then A9: lim (f2 /* seq) = lim_in+infty f2 by A2, A5, Def12; lim_in+infty f2 = lim_in+infty f2 ; then A10: f2 /* seq is convergent by A2, A5, A8, Def12; A11: rng seq c= dom f1 by A6, Lm3; then A12: lim (f1 /* seq) = lim_in+infty f1 by A1, A5, Def12; lim_in+infty f1 = lim_in+infty f1 ; then A13: f1 /* seq is convergent by A1, A5, A11, Def12; then (f1 /* seq) (#) (f2 /* seq) is convergent by A10; hence (f1 (#) f2) /* seq is convergent by A6, A7, RFUNCT_2:8; ::_thesis: lim ((f1 (#) f2) /* seq) = (lim_in+infty f1) * (lim_in+infty f2) thus lim ((f1 (#) f2) /* seq) = lim ((f1 /* seq) (#) (f2 /* seq)) by A6, A7, RFUNCT_2:8 .= (lim_in+infty f1) * (lim_in+infty f2) by A13, A12, A10, A9, SEQ_2:15 ; ::_thesis: verum end; hence f1 (#) f2 is convergent_in+infty by A3, Def6; ::_thesis: lim_in+infty (f1 (#) f2) = (lim_in+infty f1) * (lim_in+infty f2) hence lim_in+infty (f1 (#) f2) = (lim_in+infty f1) * (lim_in+infty f2) by A4, Def12; ::_thesis: verum end; theorem :: LIMFUNC1:88 for f1, f2 being PartFunc of REAL,REAL st f1 is convergent_in+infty & f2 is convergent_in+infty & lim_in+infty f2 <> 0 & ( for r being Real ex g being Real st ( r < g & g in dom (f1 / f2) ) ) holds ( f1 / f2 is convergent_in+infty & lim_in+infty (f1 / f2) = (lim_in+infty f1) / (lim_in+infty f2) ) proof let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is convergent_in+infty & f2 is convergent_in+infty & lim_in+infty f2 <> 0 & ( for r being Real ex g being Real st ( r < g & g in dom (f1 / f2) ) ) implies ( f1 / f2 is convergent_in+infty & lim_in+infty (f1 / f2) = (lim_in+infty f1) / (lim_in+infty f2) ) ) assume that A1: f1 is convergent_in+infty and A2: ( f2 is convergent_in+infty & lim_in+infty f2 <> 0 ) and A3: for r being Real ex g being Real st ( r < g & g in dom (f1 / f2) ) ; ::_thesis: ( f1 / f2 is convergent_in+infty & lim_in+infty (f1 / f2) = (lim_in+infty f1) / (lim_in+infty f2) ) dom (f1 / f2) = (dom f1) /\ ((dom f2) \ (f2 " {0})) by RFUNCT_1:def_1; then A4: dom (f1 / f2) = (dom f1) /\ (dom (f2 ^)) by RFUNCT_1:def_2; A5: (dom f1) /\ (dom (f2 ^)) c= dom (f2 ^) by XBOOLE_1:17; A6: now__::_thesis:_for_r_being_Real_ex_g_being_Real_st_ (_r_<_g_&_g_in_dom_f2_&_f2_._g_<>_0_) let r be Real; ::_thesis: ex g being Real st ( r < g & g in dom f2 & f2 . g <> 0 ) consider g being Real such that A7: r < g and A8: g in dom (f1 / f2) by A3; take g = g; ::_thesis: ( r < g & g in dom f2 & f2 . g <> 0 ) g in dom (f2 ^) by A4, A5, A8; then A9: g in (dom f2) \ (f2 " {0}) by RFUNCT_1:def_2; then ( g in dom f2 & not g in f2 " {0} ) by XBOOLE_0:def_5; then not f2 . g in {0} by FUNCT_1:def_7; hence ( r < g & g in dom f2 & f2 . g <> 0 ) by A7, A9, TARSKI:def_1, XBOOLE_0:def_5; ::_thesis: verum end; then A10: f2 ^ is convergent_in+infty by A2, Th86; A11: lim_in+infty (f2 ^) = (lim_in+infty f2) " by A2, A6, Th86; A12: now__::_thesis:_for_r_being_Real_ex_g_being_Real_st_ (_r_<_g_&_g_in_dom_(f1_(#)_(f2_^))_) let r be Real; ::_thesis: ex g being Real st ( r < g & g in dom (f1 (#) (f2 ^)) ) consider g being Real such that A13: ( r < g & g in dom (f1 / f2) ) by A3; take g = g; ::_thesis: ( r < g & g in dom (f1 (#) (f2 ^)) ) thus ( r < g & g in dom (f1 (#) (f2 ^)) ) by A4, A13, VALUED_1:def_4; ::_thesis: verum end; then f1 (#) (f2 ^) is convergent_in+infty by A1, A10, Th87; hence f1 / f2 is convergent_in+infty by RFUNCT_1:31; ::_thesis: lim_in+infty (f1 / f2) = (lim_in+infty f1) / (lim_in+infty f2) thus lim_in+infty (f1 / f2) = lim_in+infty (f1 (#) (f2 ^)) by RFUNCT_1:31 .= (lim_in+infty f1) * ((lim_in+infty f2) ") by A1, A12, A10, A11, Th87 .= (lim_in+infty f1) / (lim_in+infty f2) by XCMPLX_0:def_9 ; ::_thesis: verum end; theorem Th89: :: LIMFUNC1:89 for f being PartFunc of REAL,REAL for r being Real st f is convergent_in-infty holds ( r (#) f is convergent_in-infty & lim_in-infty (r (#) f) = r * (lim_in-infty f) ) proof let f be PartFunc of REAL,REAL; ::_thesis: for r being Real st f is convergent_in-infty holds ( r (#) f is convergent_in-infty & lim_in-infty (r (#) f) = r * (lim_in-infty f) ) let r be Real; ::_thesis: ( f is convergent_in-infty implies ( r (#) f is convergent_in-infty & lim_in-infty (r (#) f) = r * (lim_in-infty f) ) ) assume A1: f is convergent_in-infty ; ::_thesis: ( r (#) f is convergent_in-infty & lim_in-infty (r (#) f) = r * (lim_in-infty f) ) A2: now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_divergent_to-infty_&_rng_seq_c=_dom_(r_(#)_f)_holds_ (_(r_(#)_f)_/*_seq_is_convergent_&_lim_((r_(#)_f)_/*_seq)_=_r_*_(lim_in-infty_f)_) let seq be Real_Sequence; ::_thesis: ( seq is divergent_to-infty & rng seq c= dom (r (#) f) implies ( (r (#) f) /* seq is convergent & lim ((r (#) f) /* seq) = r * (lim_in-infty f) ) ) assume that A3: seq is divergent_to-infty and A4: rng seq c= dom (r (#) f) ; ::_thesis: ( (r (#) f) /* seq is convergent & lim ((r (#) f) /* seq) = r * (lim_in-infty f) ) A5: rng seq c= dom f by A4, VALUED_1:def_5; then A6: lim (f /* seq) = lim_in-infty f by A1, A3, Def13; lim_in-infty f = lim_in-infty f ; then A7: f /* seq is convergent by A1, A3, A5, Def13; then r (#) (f /* seq) is convergent ; hence (r (#) f) /* seq is convergent by A5, RFUNCT_2:9; ::_thesis: lim ((r (#) f) /* seq) = r * (lim_in-infty f) thus lim ((r (#) f) /* seq) = lim (r (#) (f /* seq)) by A5, RFUNCT_2:9 .= r * (lim_in-infty f) by A7, A6, SEQ_2:8 ; ::_thesis: verum end; for r1 being Real ex g being Real st ( g < r1 & g in dom (r (#) f) ) proof let r1 be Real; ::_thesis: ex g being Real st ( g < r1 & g in dom (r (#) f) ) consider g being Real such that A8: ( g < r1 & g in dom f ) by A1, Def9; take g ; ::_thesis: ( g < r1 & g in dom (r (#) f) ) thus ( g < r1 & g in dom (r (#) f) ) by A8, VALUED_1:def_5; ::_thesis: verum end; hence r (#) f is convergent_in-infty by A2, Def9; ::_thesis: lim_in-infty (r (#) f) = r * (lim_in-infty f) hence lim_in-infty (r (#) f) = r * (lim_in-infty f) by A2, Def13; ::_thesis: verum end; theorem Th90: :: LIMFUNC1:90 for f being PartFunc of REAL,REAL st f is convergent_in-infty holds ( - f is convergent_in-infty & lim_in-infty (- f) = - (lim_in-infty f) ) proof let f be PartFunc of REAL,REAL; ::_thesis: ( f is convergent_in-infty implies ( - f is convergent_in-infty & lim_in-infty (- f) = - (lim_in-infty f) ) ) assume A1: f is convergent_in-infty ; ::_thesis: ( - f is convergent_in-infty & lim_in-infty (- f) = - (lim_in-infty f) ) (- 1) (#) f = - f ; hence - f is convergent_in-infty by A1, Th89; ::_thesis: lim_in-infty (- f) = - (lim_in-infty f) thus lim_in-infty (- f) = (- 1) * (lim_in-infty f) by A1, Th89 .= - (lim_in-infty f) ; ::_thesis: verum end; theorem Th91: :: LIMFUNC1:91 for f1, f2 being PartFunc of REAL,REAL st f1 is convergent_in-infty & f2 is convergent_in-infty & ( for r being Real ex g being Real st ( g < r & g in dom (f1 + f2) ) ) holds ( f1 + f2 is convergent_in-infty & lim_in-infty (f1 + f2) = (lim_in-infty f1) + (lim_in-infty f2) ) proof let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is convergent_in-infty & f2 is convergent_in-infty & ( for r being Real ex g being Real st ( g < r & g in dom (f1 + f2) ) ) implies ( f1 + f2 is convergent_in-infty & lim_in-infty (f1 + f2) = (lim_in-infty f1) + (lim_in-infty f2) ) ) assume that A1: f1 is convergent_in-infty and A2: f2 is convergent_in-infty and A3: for r being Real ex g being Real st ( g < r & g in dom (f1 + f2) ) ; ::_thesis: ( f1 + f2 is convergent_in-infty & lim_in-infty (f1 + f2) = (lim_in-infty f1) + (lim_in-infty f2) ) A4: now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_divergent_to-infty_&_rng_seq_c=_dom_(f1_+_f2)_holds_ (_(f1_+_f2)_/*_seq_is_convergent_&_lim_((f1_+_f2)_/*_seq)_=_(lim_in-infty_f1)_+_(lim_in-infty_f2)_) let seq be Real_Sequence; ::_thesis: ( seq is divergent_to-infty & rng seq c= dom (f1 + f2) implies ( (f1 + f2) /* seq is convergent & lim ((f1 + f2) /* seq) = (lim_in-infty f1) + (lim_in-infty f2) ) ) assume that A5: seq is divergent_to-infty and A6: rng seq c= dom (f1 + f2) ; ::_thesis: ( (f1 + f2) /* seq is convergent & lim ((f1 + f2) /* seq) = (lim_in-infty f1) + (lim_in-infty f2) ) A7: rng seq c= (dom f1) /\ (dom f2) by A6, VALUED_1:def_1; (dom f1) /\ (dom f2) c= dom f2 by XBOOLE_1:17; then A8: rng seq c= dom f2 by A7, XBOOLE_1:1; then A9: lim (f2 /* seq) = lim_in-infty f2 by A2, A5, Def13; lim_in-infty f2 = lim_in-infty f2 ; then A10: f2 /* seq is convergent by A2, A5, A8, Def13; (dom f1) /\ (dom f2) c= dom f1 by XBOOLE_1:17; then A11: rng seq c= dom f1 by A7, XBOOLE_1:1; then A12: lim (f1 /* seq) = lim_in-infty f1 by A1, A5, Def13; lim_in-infty f1 = lim_in-infty f1 ; then A13: f1 /* seq is convergent by A1, A5, A11, Def13; then (f1 /* seq) + (f2 /* seq) is convergent by A10; hence (f1 + f2) /* seq is convergent by A7, RFUNCT_2:8; ::_thesis: lim ((f1 + f2) /* seq) = (lim_in-infty f1) + (lim_in-infty f2) thus lim ((f1 + f2) /* seq) = lim ((f1 /* seq) + (f2 /* seq)) by A7, RFUNCT_2:8 .= (lim_in-infty f1) + (lim_in-infty f2) by A13, A12, A10, A9, SEQ_2:6 ; ::_thesis: verum end; hence f1 + f2 is convergent_in-infty by A3, Def9; ::_thesis: lim_in-infty (f1 + f2) = (lim_in-infty f1) + (lim_in-infty f2) hence lim_in-infty (f1 + f2) = (lim_in-infty f1) + (lim_in-infty f2) by A4, Def13; ::_thesis: verum end; theorem :: LIMFUNC1:92 for f1, f2 being PartFunc of REAL,REAL st f1 is convergent_in-infty & f2 is convergent_in-infty & ( for r being Real ex g being Real st ( g < r & g in dom (f1 - f2) ) ) holds ( f1 - f2 is convergent_in-infty & lim_in-infty (f1 - f2) = (lim_in-infty f1) - (lim_in-infty f2) ) proof let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is convergent_in-infty & f2 is convergent_in-infty & ( for r being Real ex g being Real st ( g < r & g in dom (f1 - f2) ) ) implies ( f1 - f2 is convergent_in-infty & lim_in-infty (f1 - f2) = (lim_in-infty f1) - (lim_in-infty f2) ) ) assume that A1: f1 is convergent_in-infty and A2: f2 is convergent_in-infty and A3: for r being Real ex g being Real st ( g < r & g in dom (f1 - f2) ) ; ::_thesis: ( f1 - f2 is convergent_in-infty & lim_in-infty (f1 - f2) = (lim_in-infty f1) - (lim_in-infty f2) ) A4: - f2 is convergent_in-infty by A2, Th90; hence f1 - f2 is convergent_in-infty by A1, A3, Th91; ::_thesis: lim_in-infty (f1 - f2) = (lim_in-infty f1) - (lim_in-infty f2) lim_in-infty (- f2) = - (lim_in-infty f2) by A2, Th90; hence lim_in-infty (f1 - f2) = (lim_in-infty f1) + (- (lim_in-infty f2)) by A1, A3, A4, Th91 .= (lim_in-infty f1) - (lim_in-infty f2) ; ::_thesis: verum end; theorem :: LIMFUNC1:93 for f being PartFunc of REAL,REAL st f is convergent_in-infty & f " {0} = {} & lim_in-infty f <> 0 holds ( f ^ is convergent_in-infty & lim_in-infty (f ^) = (lim_in-infty f) " ) proof let f be PartFunc of REAL,REAL; ::_thesis: ( f is convergent_in-infty & f " {0} = {} & lim_in-infty f <> 0 implies ( f ^ is convergent_in-infty & lim_in-infty (f ^) = (lim_in-infty f) " ) ) assume that A1: f is convergent_in-infty and A2: f " {0} = {} and A3: lim_in-infty f <> 0 ; ::_thesis: ( f ^ is convergent_in-infty & lim_in-infty (f ^) = (lim_in-infty f) " ) A4: dom f = (dom f) \ (f " {0}) by A2 .= dom (f ^) by RFUNCT_1:def_2 ; A5: now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_divergent_to-infty_&_rng_seq_c=_dom_(f_^)_holds_ (_(f_^)_/*_seq_is_convergent_&_lim_((f_^)_/*_seq)_=_(lim_in-infty_f)_"_) let seq be Real_Sequence; ::_thesis: ( seq is divergent_to-infty & rng seq c= dom (f ^) implies ( (f ^) /* seq is convergent & lim ((f ^) /* seq) = (lim_in-infty f) " ) ) assume that A6: seq is divergent_to-infty and A7: rng seq c= dom (f ^) ; ::_thesis: ( (f ^) /* seq is convergent & lim ((f ^) /* seq) = (lim_in-infty f) " ) A8: ( f /* seq is convergent & lim (f /* seq) = lim_in-infty f ) by A1, A3, A4, A6, A7, Def13; then (f /* seq) " is convergent by A3, A7, RFUNCT_2:11, SEQ_2:21; hence (f ^) /* seq is convergent by A7, RFUNCT_2:12; ::_thesis: lim ((f ^) /* seq) = (lim_in-infty f) " thus lim ((f ^) /* seq) = lim ((f /* seq) ") by A7, RFUNCT_2:12 .= (lim_in-infty f) " by A3, A7, A8, RFUNCT_2:11, SEQ_2:22 ; ::_thesis: verum end; for r being Real ex g being Real st ( g < r & g in dom (f ^) ) by A1, A4, Def9; hence f ^ is convergent_in-infty by A5, Def9; ::_thesis: lim_in-infty (f ^) = (lim_in-infty f) " hence lim_in-infty (f ^) = (lim_in-infty f) " by A5, Def13; ::_thesis: verum end; theorem :: LIMFUNC1:94 for f being PartFunc of REAL,REAL st f is convergent_in-infty holds ( abs f is convergent_in-infty & lim_in-infty (abs f) = abs (lim_in-infty f) ) proof let f be PartFunc of REAL,REAL; ::_thesis: ( f is convergent_in-infty implies ( abs f is convergent_in-infty & lim_in-infty (abs f) = abs (lim_in-infty f) ) ) assume A1: f is convergent_in-infty ; ::_thesis: ( abs f is convergent_in-infty & lim_in-infty (abs f) = abs (lim_in-infty f) ) A2: now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_divergent_to-infty_&_rng_seq_c=_dom_(abs_f)_holds_ (_(abs_f)_/*_seq_is_convergent_&_lim_((abs_f)_/*_seq)_=_abs_(lim_in-infty_f)_) let seq be Real_Sequence; ::_thesis: ( seq is divergent_to-infty & rng seq c= dom (abs f) implies ( (abs f) /* seq is convergent & lim ((abs f) /* seq) = abs (lim_in-infty f) ) ) assume that A3: seq is divergent_to-infty and A4: rng seq c= dom (abs f) ; ::_thesis: ( (abs f) /* seq is convergent & lim ((abs f) /* seq) = abs (lim_in-infty f) ) A5: rng seq c= dom f by A4, VALUED_1:def_11; then A6: lim (f /* seq) = lim_in-infty f by A1, A3, Def13; lim_in-infty f = lim_in-infty f ; then A7: f /* seq is convergent by A1, A3, A5, Def13; then abs (f /* seq) is convergent ; hence (abs f) /* seq is convergent by A5, RFUNCT_2:10; ::_thesis: lim ((abs f) /* seq) = abs (lim_in-infty f) thus lim ((abs f) /* seq) = lim (abs (f /* seq)) by A5, RFUNCT_2:10 .= abs (lim_in-infty f) by A7, A6, SEQ_4:14 ; ::_thesis: verum end; now__::_thesis:_for_r_being_Real_ex_g_being_Real_st_ (_g_<_r_&_g_in_dom_(abs_f)_) let r be Real; ::_thesis: ex g being Real st ( g < r & g in dom (abs f) ) consider g being Real such that A8: ( g < r & g in dom f ) by A1, Def9; take g = g; ::_thesis: ( g < r & g in dom (abs f) ) thus ( g < r & g in dom (abs f) ) by A8, VALUED_1:def_11; ::_thesis: verum end; hence abs f is convergent_in-infty by A2, Def9; ::_thesis: lim_in-infty (abs f) = abs (lim_in-infty f) hence lim_in-infty (abs f) = abs (lim_in-infty f) by A2, Def13; ::_thesis: verum end; theorem Th95: :: LIMFUNC1:95 for f being PartFunc of REAL,REAL st f is convergent_in-infty & lim_in-infty f <> 0 & ( for r being Real ex g being Real st ( g < r & g in dom f & f . g <> 0 ) ) holds ( f ^ is convergent_in-infty & lim_in-infty (f ^) = (lim_in-infty f) " ) proof let f be PartFunc of REAL,REAL; ::_thesis: ( f is convergent_in-infty & lim_in-infty f <> 0 & ( for r being Real ex g being Real st ( g < r & g in dom f & f . g <> 0 ) ) implies ( f ^ is convergent_in-infty & lim_in-infty (f ^) = (lim_in-infty f) " ) ) assume that A1: f is convergent_in-infty and A2: lim_in-infty f <> 0 and A3: for r being Real ex g being Real st ( g < r & g in dom f & f . g <> 0 ) ; ::_thesis: ( f ^ is convergent_in-infty & lim_in-infty (f ^) = (lim_in-infty f) " ) A4: now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_divergent_to-infty_&_rng_seq_c=_dom_(f_^)_holds_ (_(f_^)_/*_seq_is_convergent_&_lim_((f_^)_/*_seq)_=_(lim_in-infty_f)_"_) let seq be Real_Sequence; ::_thesis: ( seq is divergent_to-infty & rng seq c= dom (f ^) implies ( (f ^) /* seq is convergent & lim ((f ^) /* seq) = (lim_in-infty f) " ) ) assume that A5: seq is divergent_to-infty and A6: rng seq c= dom (f ^) ; ::_thesis: ( (f ^) /* seq is convergent & lim ((f ^) /* seq) = (lim_in-infty f) " ) ( dom (f ^) = (dom f) \ (f " {0}) & (dom f) \ (f " {0}) c= dom f ) by RFUNCT_1:def_2, XBOOLE_1:36; then rng seq c= dom f by A6, XBOOLE_1:1; then A7: ( f /* seq is convergent & lim (f /* seq) = lim_in-infty f ) by A1, A2, A5, Def13; then (f /* seq) " is convergent by A2, A6, RFUNCT_2:11, SEQ_2:21; hence (f ^) /* seq is convergent by A6, RFUNCT_2:12; ::_thesis: lim ((f ^) /* seq) = (lim_in-infty f) " thus lim ((f ^) /* seq) = lim ((f /* seq) ") by A6, RFUNCT_2:12 .= (lim_in-infty f) " by A2, A6, A7, RFUNCT_2:11, SEQ_2:22 ; ::_thesis: verum end; now__::_thesis:_for_r_being_Real_ex_g_being_Real_st_ (_g_<_r_&_g_in_dom_(f_^)_) let r be Real; ::_thesis: ex g being Real st ( g < r & g in dom (f ^) ) consider g being Real such that A8: g < r and A9: g in dom f and A10: f . g <> 0 by A3; take g = g; ::_thesis: ( g < r & g in dom (f ^) ) not f . g in {0} by A10, TARSKI:def_1; then not g in f " {0} by FUNCT_1:def_7; then g in (dom f) \ (f " {0}) by A9, XBOOLE_0:def_5; hence ( g < r & g in dom (f ^) ) by A8, RFUNCT_1:def_2; ::_thesis: verum end; hence f ^ is convergent_in-infty by A4, Def9; ::_thesis: lim_in-infty (f ^) = (lim_in-infty f) " hence lim_in-infty (f ^) = (lim_in-infty f) " by A4, Def13; ::_thesis: verum end; theorem Th96: :: LIMFUNC1:96 for f1, f2 being PartFunc of REAL,REAL st f1 is convergent_in-infty & f2 is convergent_in-infty & ( for r being Real ex g being Real st ( g < r & g in dom (f1 (#) f2) ) ) holds ( f1 (#) f2 is convergent_in-infty & lim_in-infty (f1 (#) f2) = (lim_in-infty f1) * (lim_in-infty f2) ) proof let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is convergent_in-infty & f2 is convergent_in-infty & ( for r being Real ex g being Real st ( g < r & g in dom (f1 (#) f2) ) ) implies ( f1 (#) f2 is convergent_in-infty & lim_in-infty (f1 (#) f2) = (lim_in-infty f1) * (lim_in-infty f2) ) ) assume that A1: f1 is convergent_in-infty and A2: f2 is convergent_in-infty and A3: for r being Real ex g being Real st ( g < r & g in dom (f1 (#) f2) ) ; ::_thesis: ( f1 (#) f2 is convergent_in-infty & lim_in-infty (f1 (#) f2) = (lim_in-infty f1) * (lim_in-infty f2) ) A4: now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_divergent_to-infty_&_rng_seq_c=_dom_(f1_(#)_f2)_holds_ (_(f1_(#)_f2)_/*_seq_is_convergent_&_lim_((f1_(#)_f2)_/*_seq)_=_(lim_in-infty_f1)_*_(lim_in-infty_f2)_) let seq be Real_Sequence; ::_thesis: ( seq is divergent_to-infty & rng seq c= dom (f1 (#) f2) implies ( (f1 (#) f2) /* seq is convergent & lim ((f1 (#) f2) /* seq) = (lim_in-infty f1) * (lim_in-infty f2) ) ) assume that A5: seq is divergent_to-infty and A6: rng seq c= dom (f1 (#) f2) ; ::_thesis: ( (f1 (#) f2) /* seq is convergent & lim ((f1 (#) f2) /* seq) = (lim_in-infty f1) * (lim_in-infty f2) ) A7: dom (f1 (#) f2) = (dom f1) /\ (dom f2) by VALUED_1:def_4; (dom f1) /\ (dom f2) c= dom f2 by XBOOLE_1:17; then A8: rng seq c= dom f2 by A6, A7, XBOOLE_1:1; then A9: lim (f2 /* seq) = lim_in-infty f2 by A2, A5, Def13; lim_in-infty f2 = lim_in-infty f2 ; then A10: f2 /* seq is convergent by A2, A5, A8, Def13; (dom f1) /\ (dom f2) c= dom f1 by XBOOLE_1:17; then A11: rng seq c= dom f1 by A6, A7, XBOOLE_1:1; then A12: lim (f1 /* seq) = lim_in-infty f1 by A1, A5, Def13; lim_in-infty f1 = lim_in-infty f1 ; then A13: f1 /* seq is convergent by A1, A5, A11, Def13; then (f1 /* seq) (#) (f2 /* seq) is convergent by A10; hence (f1 (#) f2) /* seq is convergent by A6, A7, RFUNCT_2:8; ::_thesis: lim ((f1 (#) f2) /* seq) = (lim_in-infty f1) * (lim_in-infty f2) thus lim ((f1 (#) f2) /* seq) = lim ((f1 /* seq) (#) (f2 /* seq)) by A6, A7, RFUNCT_2:8 .= (lim_in-infty f1) * (lim_in-infty f2) by A13, A12, A10, A9, SEQ_2:15 ; ::_thesis: verum end; hence f1 (#) f2 is convergent_in-infty by A3, Def9; ::_thesis: lim_in-infty (f1 (#) f2) = (lim_in-infty f1) * (lim_in-infty f2) hence lim_in-infty (f1 (#) f2) = (lim_in-infty f1) * (lim_in-infty f2) by A4, Def13; ::_thesis: verum end; theorem :: LIMFUNC1:97 for f1, f2 being PartFunc of REAL,REAL st f1 is convergent_in-infty & f2 is convergent_in-infty & lim_in-infty f2 <> 0 & ( for r being Real ex g being Real st ( g < r & g in dom (f1 / f2) ) ) holds ( f1 / f2 is convergent_in-infty & lim_in-infty (f1 / f2) = (lim_in-infty f1) / (lim_in-infty f2) ) proof let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is convergent_in-infty & f2 is convergent_in-infty & lim_in-infty f2 <> 0 & ( for r being Real ex g being Real st ( g < r & g in dom (f1 / f2) ) ) implies ( f1 / f2 is convergent_in-infty & lim_in-infty (f1 / f2) = (lim_in-infty f1) / (lim_in-infty f2) ) ) assume that A1: f1 is convergent_in-infty and A2: ( f2 is convergent_in-infty & lim_in-infty f2 <> 0 ) and A3: for r being Real ex g being Real st ( g < r & g in dom (f1 / f2) ) ; ::_thesis: ( f1 / f2 is convergent_in-infty & lim_in-infty (f1 / f2) = (lim_in-infty f1) / (lim_in-infty f2) ) dom (f1 / f2) = (dom f1) /\ ((dom f2) \ (f2 " {0})) by RFUNCT_1:def_1; then A4: dom (f1 / f2) = (dom f1) /\ (dom (f2 ^)) by RFUNCT_1:def_2; A5: (dom f1) /\ (dom (f2 ^)) c= dom (f2 ^) by XBOOLE_1:17; A6: now__::_thesis:_for_r_being_Real_ex_g_being_Real_st_ (_g_<_r_&_g_in_dom_f2_&_f2_._g_<>_0_) let r be Real; ::_thesis: ex g being Real st ( g < r & g in dom f2 & f2 . g <> 0 ) consider g being Real such that A7: g < r and A8: g in dom (f1 / f2) by A3; take g = g; ::_thesis: ( g < r & g in dom f2 & f2 . g <> 0 ) g in dom (f2 ^) by A4, A5, A8; then A9: g in (dom f2) \ (f2 " {0}) by RFUNCT_1:def_2; then ( g in dom f2 & not g in f2 " {0} ) by XBOOLE_0:def_5; then not f2 . g in {0} by FUNCT_1:def_7; hence ( g < r & g in dom f2 & f2 . g <> 0 ) by A7, A9, TARSKI:def_1, XBOOLE_0:def_5; ::_thesis: verum end; then A10: f2 ^ is convergent_in-infty by A2, Th95; A11: lim_in-infty (f2 ^) = (lim_in-infty f2) " by A2, A6, Th95; A12: now__::_thesis:_for_r_being_Real_ex_g_being_Real_st_ (_g_<_r_&_g_in_dom_(f1_(#)_(f2_^))_) let r be Real; ::_thesis: ex g being Real st ( g < r & g in dom (f1 (#) (f2 ^)) ) consider g being Real such that A13: ( g < r & g in dom (f1 / f2) ) by A3; take g = g; ::_thesis: ( g < r & g in dom (f1 (#) (f2 ^)) ) thus ( g < r & g in dom (f1 (#) (f2 ^)) ) by A4, A13, VALUED_1:def_4; ::_thesis: verum end; then f1 (#) (f2 ^) is convergent_in-infty by A1, A10, Th96; hence f1 / f2 is convergent_in-infty by RFUNCT_1:31; ::_thesis: lim_in-infty (f1 / f2) = (lim_in-infty f1) / (lim_in-infty f2) thus lim_in-infty (f1 / f2) = lim_in-infty (f1 (#) (f2 ^)) by RFUNCT_1:31 .= (lim_in-infty f1) * ((lim_in-infty f2) ") by A1, A12, A10, A11, Th96 .= (lim_in-infty f1) / (lim_in-infty f2) by XCMPLX_0:def_9 ; ::_thesis: verum end; theorem :: LIMFUNC1:98 for f1, f2 being PartFunc of REAL,REAL st f1 is convergent_in+infty & lim_in+infty f1 = 0 & ( for r being Real ex g being Real st ( r < g & g in dom (f1 (#) f2) ) ) & ex r being Real st f2 | (right_open_halfline r) is bounded holds ( f1 (#) f2 is convergent_in+infty & lim_in+infty (f1 (#) f2) = 0 ) proof let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is convergent_in+infty & lim_in+infty f1 = 0 & ( for r being Real ex g being Real st ( r < g & g in dom (f1 (#) f2) ) ) & ex r being Real st f2 | (right_open_halfline r) is bounded implies ( f1 (#) f2 is convergent_in+infty & lim_in+infty (f1 (#) f2) = 0 ) ) assume that A1: ( f1 is convergent_in+infty & lim_in+infty f1 = 0 ) and A2: for r being Real ex g being Real st ( r < g & g in dom (f1 (#) f2) ) ; ::_thesis: ( for r being Real holds not f2 | (right_open_halfline r) is bounded or ( f1 (#) f2 is convergent_in+infty & lim_in+infty (f1 (#) f2) = 0 ) ) given r being Real such that A3: f2 | (right_open_halfline r) is bounded ; ::_thesis: ( f1 (#) f2 is convergent_in+infty & lim_in+infty (f1 (#) f2) = 0 ) consider g being real number such that A4: for r1 being set st r1 in (right_open_halfline r) /\ (dom f2) holds abs (f2 . r1) <= g by A3, RFUNCT_1:73; A5: now__::_thesis:_for_s_being_Real_Sequence_st_s_is_divergent_to+infty_&_rng_s_c=_dom_(f1_(#)_f2)_holds_ (_(f1_(#)_f2)_/*_s_is_convergent_&_lim_((f1_(#)_f2)_/*_s)_=_0_) let s be Real_Sequence; ::_thesis: ( s is divergent_to+infty & rng s c= dom (f1 (#) f2) implies ( (f1 (#) f2) /* s is convergent & lim ((f1 (#) f2) /* s) = 0 ) ) assume that A6: s is divergent_to+infty and A7: rng s c= dom (f1 (#) f2) ; ::_thesis: ( (f1 (#) f2) /* s is convergent & lim ((f1 (#) f2) /* s) = 0 ) consider k being Element of NAT such that A8: for n being Element of NAT st k <= n holds r < s . n by A6, Def4; A9: rng (s ^\ k) c= rng s by VALUED_0:21; A10: rng s c= dom f2 by A7, Lm3; then A11: rng (s ^\ k) c= dom f2 by A9, XBOOLE_1:1; now__::_thesis:_(_0_<_(abs_g)_+_1_&_(_for_n_being_Element_of_NAT_holds_abs_((f2_/*_(s_^\_k))_._n)_<_(abs_g)_+_1_)_) set t = (abs g) + 1; 0 <= abs g by COMPLEX1:46; hence 0 < (abs g) + 1 ; ::_thesis: for n being Element of NAT holds abs ((f2 /* (s ^\ k)) . n) < (abs g) + 1 let n be Element of NAT ; ::_thesis: abs ((f2 /* (s ^\ k)) . n) < (abs g) + 1 r < s . (n + k) by A8, NAT_1:12; then r < (s ^\ k) . n by NAT_1:def_3; then (s ^\ k) . n in { g1 where g1 is Real : r < g1 } ; then ( (s ^\ k) . n in rng (s ^\ k) & (s ^\ k) . n in right_open_halfline r ) by VALUED_0:28, XXREAL_1:230; then (s ^\ k) . n in (right_open_halfline r) /\ (dom f2) by A11, XBOOLE_0:def_4; then abs (f2 . ((s ^\ k) . n)) <= g by A4; then A12: abs ((f2 /* (s ^\ k)) . n) <= g by A10, A9, FUNCT_2:108, XBOOLE_1:1; g <= abs g by ABSVALUE:4; then g < (abs g) + 1 by Lm1; hence abs ((f2 /* (s ^\ k)) . n) < (abs g) + 1 by A12, XXREAL_0:2; ::_thesis: verum end; then A13: f2 /* (s ^\ k) is bounded by SEQ_2:3; dom (f1 (#) f2) = (dom f1) /\ (dom f2) by A7, Lm3; then rng (s ^\ k) c= (dom f1) /\ (dom f2) by A7, A9, XBOOLE_1:1; then A14: (f1 /* (s ^\ k)) (#) (f2 /* (s ^\ k)) = (f1 (#) f2) /* (s ^\ k) by RFUNCT_2:8 .= ((f1 (#) f2) /* s) ^\ k by A7, VALUED_0:27 ; rng s c= dom f1 by A7, Lm3; then A15: rng (s ^\ k) c= dom f1 by A9, XBOOLE_1:1; s ^\ k is divergent_to+infty by A6, Th26; then A16: ( f1 /* (s ^\ k) is convergent & lim (f1 /* (s ^\ k)) = 0 ) by A1, A15, Def12; then A17: (f1 /* (s ^\ k)) (#) (f2 /* (s ^\ k)) is convergent by A13, SEQ_2:25; hence (f1 (#) f2) /* s is convergent by A14, SEQ_4:21; ::_thesis: lim ((f1 (#) f2) /* s) = 0 lim ((f1 /* (s ^\ k)) (#) (f2 /* (s ^\ k))) = 0 by A16, A13, SEQ_2:26; hence lim ((f1 (#) f2) /* s) = 0 by A17, A14, SEQ_4:22; ::_thesis: verum end; hence f1 (#) f2 is convergent_in+infty by A2, Def6; ::_thesis: lim_in+infty (f1 (#) f2) = 0 hence lim_in+infty (f1 (#) f2) = 0 by A5, Def12; ::_thesis: verum end; theorem :: LIMFUNC1:99 for f1, f2 being PartFunc of REAL,REAL st f1 is convergent_in-infty & lim_in-infty f1 = 0 & ( for r being Real ex g being Real st ( g < r & g in dom (f1 (#) f2) ) ) & ex r being Real st f2 | (left_open_halfline r) is bounded holds ( f1 (#) f2 is convergent_in-infty & lim_in-infty (f1 (#) f2) = 0 ) proof let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is convergent_in-infty & lim_in-infty f1 = 0 & ( for r being Real ex g being Real st ( g < r & g in dom (f1 (#) f2) ) ) & ex r being Real st f2 | (left_open_halfline r) is bounded implies ( f1 (#) f2 is convergent_in-infty & lim_in-infty (f1 (#) f2) = 0 ) ) assume that A1: ( f1 is convergent_in-infty & lim_in-infty f1 = 0 ) and A2: for r being Real ex g being Real st ( g < r & g in dom (f1 (#) f2) ) ; ::_thesis: ( for r being Real holds not f2 | (left_open_halfline r) is bounded or ( f1 (#) f2 is convergent_in-infty & lim_in-infty (f1 (#) f2) = 0 ) ) given r being Real such that A3: f2 | (left_open_halfline r) is bounded ; ::_thesis: ( f1 (#) f2 is convergent_in-infty & lim_in-infty (f1 (#) f2) = 0 ) consider g being real number such that A4: for r1 being set st r1 in (left_open_halfline r) /\ (dom f2) holds abs (f2 . r1) <= g by A3, RFUNCT_1:73; A5: now__::_thesis:_for_s_being_Real_Sequence_st_s_is_divergent_to-infty_&_rng_s_c=_dom_(f1_(#)_f2)_holds_ (_(f1_(#)_f2)_/*_s_is_convergent_&_lim_((f1_(#)_f2)_/*_s)_=_0_) let s be Real_Sequence; ::_thesis: ( s is divergent_to-infty & rng s c= dom (f1 (#) f2) implies ( (f1 (#) f2) /* s is convergent & lim ((f1 (#) f2) /* s) = 0 ) ) assume that A6: s is divergent_to-infty and A7: rng s c= dom (f1 (#) f2) ; ::_thesis: ( (f1 (#) f2) /* s is convergent & lim ((f1 (#) f2) /* s) = 0 ) consider k being Element of NAT such that A8: for n being Element of NAT st k <= n holds s . n < r by A6, Def5; A9: rng (s ^\ k) c= rng s by VALUED_0:21; A10: rng s c= dom f2 by A7, Lm3; then A11: rng (s ^\ k) c= dom f2 by A9, XBOOLE_1:1; now__::_thesis:_(_0_<_(abs_g)_+_1_&_(_for_n_being_Element_of_NAT_holds_abs_((f2_/*_(s_^\_k))_._n)_<_(abs_g)_+_1_)_) set t = (abs g) + 1; 0 <= abs g by COMPLEX1:46; hence 0 < (abs g) + 1 ; ::_thesis: for n being Element of NAT holds abs ((f2 /* (s ^\ k)) . n) < (abs g) + 1 let n be Element of NAT ; ::_thesis: abs ((f2 /* (s ^\ k)) . n) < (abs g) + 1 s . (n + k) < r by A8, NAT_1:12; then (s ^\ k) . n < r by NAT_1:def_3; then (s ^\ k) . n in { g1 where g1 is Real : g1 < r } ; then ( (s ^\ k) . n in rng (s ^\ k) & (s ^\ k) . n in left_open_halfline r ) by VALUED_0:28, XXREAL_1:229; then (s ^\ k) . n in (left_open_halfline r) /\ (dom f2) by A11, XBOOLE_0:def_4; then abs (f2 . ((s ^\ k) . n)) <= g by A4; then A12: abs ((f2 /* (s ^\ k)) . n) <= g by A10, A9, FUNCT_2:108, XBOOLE_1:1; g <= abs g by ABSVALUE:4; then g < (abs g) + 1 by Lm1; hence abs ((f2 /* (s ^\ k)) . n) < (abs g) + 1 by A12, XXREAL_0:2; ::_thesis: verum end; then A13: f2 /* (s ^\ k) is bounded by SEQ_2:3; dom (f1 (#) f2) = (dom f1) /\ (dom f2) by A7, Lm3; then rng (s ^\ k) c= (dom f1) /\ (dom f2) by A7, A9, XBOOLE_1:1; then A14: (f1 /* (s ^\ k)) (#) (f2 /* (s ^\ k)) = (f1 (#) f2) /* (s ^\ k) by RFUNCT_2:8 .= ((f1 (#) f2) /* s) ^\ k by A7, VALUED_0:27 ; rng s c= dom f1 by A7, Lm3; then A15: rng (s ^\ k) c= dom f1 by A9, XBOOLE_1:1; s ^\ k is divergent_to-infty by A6, Th27; then A16: ( f1 /* (s ^\ k) is convergent & lim (f1 /* (s ^\ k)) = 0 ) by A1, A15, Def13; then A17: (f1 /* (s ^\ k)) (#) (f2 /* (s ^\ k)) is convergent by A13, SEQ_2:25; hence (f1 (#) f2) /* s is convergent by A14, SEQ_4:21; ::_thesis: lim ((f1 (#) f2) /* s) = 0 lim ((f1 /* (s ^\ k)) (#) (f2 /* (s ^\ k))) = 0 by A16, A13, SEQ_2:26; hence lim ((f1 (#) f2) /* s) = 0 by A17, A14, SEQ_4:22; ::_thesis: verum end; hence f1 (#) f2 is convergent_in-infty by A2, Def9; ::_thesis: lim_in-infty (f1 (#) f2) = 0 hence lim_in-infty (f1 (#) f2) = 0 by A5, Def13; ::_thesis: verum end; theorem Th100: :: LIMFUNC1:100 for f1, f2, f being PartFunc of REAL,REAL st f1 is convergent_in+infty & f2 is convergent_in+infty & lim_in+infty f1 = lim_in+infty f2 & ( for r being Real ex g being Real st ( r < g & g in dom f ) ) & ex r being Real st ( ( ( (dom f1) /\ (right_open_halfline r) c= (dom f2) /\ (right_open_halfline r) & (dom f) /\ (right_open_halfline r) c= (dom f1) /\ (right_open_halfline r) ) or ( (dom f2) /\ (right_open_halfline r) c= (dom f1) /\ (right_open_halfline r) & (dom f) /\ (right_open_halfline r) c= (dom f2) /\ (right_open_halfline r) ) ) & ( for g being Real st g in (dom f) /\ (right_open_halfline r) holds ( f1 . g <= f . g & f . g <= f2 . g ) ) ) holds ( f is convergent_in+infty & lim_in+infty f = lim_in+infty f1 ) proof let f1, f2, f be PartFunc of REAL,REAL; ::_thesis: ( f1 is convergent_in+infty & f2 is convergent_in+infty & lim_in+infty f1 = lim_in+infty f2 & ( for r being Real ex g being Real st ( r < g & g in dom f ) ) & ex r being Real st ( ( ( (dom f1) /\ (right_open_halfline r) c= (dom f2) /\ (right_open_halfline r) & (dom f) /\ (right_open_halfline r) c= (dom f1) /\ (right_open_halfline r) ) or ( (dom f2) /\ (right_open_halfline r) c= (dom f1) /\ (right_open_halfline r) & (dom f) /\ (right_open_halfline r) c= (dom f2) /\ (right_open_halfline r) ) ) & ( for g being Real st g in (dom f) /\ (right_open_halfline r) holds ( f1 . g <= f . g & f . g <= f2 . g ) ) ) implies ( f is convergent_in+infty & lim_in+infty f = lim_in+infty f1 ) ) assume that A1: f1 is convergent_in+infty and A2: f2 is convergent_in+infty and A3: lim_in+infty f1 = lim_in+infty f2 and A4: for r being Real ex g being Real st ( r < g & g in dom f ) ; ::_thesis: ( for r being Real holds ( ( not ( (dom f1) /\ (right_open_halfline r) c= (dom f2) /\ (right_open_halfline r) & (dom f) /\ (right_open_halfline r) c= (dom f1) /\ (right_open_halfline r) ) & not ( (dom f2) /\ (right_open_halfline r) c= (dom f1) /\ (right_open_halfline r) & (dom f) /\ (right_open_halfline r) c= (dom f2) /\ (right_open_halfline r) ) ) or ex g being Real st ( g in (dom f) /\ (right_open_halfline r) & not ( f1 . g <= f . g & f . g <= f2 . g ) ) ) or ( f is convergent_in+infty & lim_in+infty f = lim_in+infty f1 ) ) given r1 being Real such that A5: ( ( (dom f1) /\ (right_open_halfline r1) c= (dom f2) /\ (right_open_halfline r1) & (dom f) /\ (right_open_halfline r1) c= (dom f1) /\ (right_open_halfline r1) ) or ( (dom f2) /\ (right_open_halfline r1) c= (dom f1) /\ (right_open_halfline r1) & (dom f) /\ (right_open_halfline r1) c= (dom f2) /\ (right_open_halfline r1) ) ) and A6: for g being Real st g in (dom f) /\ (right_open_halfline r1) holds ( f1 . g <= f . g & f . g <= f2 . g ) ; ::_thesis: ( f is convergent_in+infty & lim_in+infty f = lim_in+infty f1 ) now__::_thesis:_(_f_is_convergent_in+infty_&_f_is_convergent_in+infty_&_lim_in+infty_f_=_lim_in+infty_f1_) percases ( ( (dom f1) /\ (right_open_halfline r1) c= (dom f2) /\ (right_open_halfline r1) & (dom f) /\ (right_open_halfline r1) c= (dom f1) /\ (right_open_halfline r1) ) or ( (dom f2) /\ (right_open_halfline r1) c= (dom f1) /\ (right_open_halfline r1) & (dom f) /\ (right_open_halfline r1) c= (dom f2) /\ (right_open_halfline r1) ) ) by A5; supposeA7: ( (dom f1) /\ (right_open_halfline r1) c= (dom f2) /\ (right_open_halfline r1) & (dom f) /\ (right_open_halfline r1) c= (dom f1) /\ (right_open_halfline r1) ) ; ::_thesis: ( f is convergent_in+infty & f is convergent_in+infty & lim_in+infty f = lim_in+infty f1 ) A8: now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_divergent_to+infty_&_rng_seq_c=_dom_f_holds_ (_f_/*_seq_is_convergent_&_lim_(f_/*_seq)_=_lim_in+infty_f1_) let seq be Real_Sequence; ::_thesis: ( seq is divergent_to+infty & rng seq c= dom f implies ( f /* seq is convergent & lim (f /* seq) = lim_in+infty f1 ) ) assume that A9: seq is divergent_to+infty and A10: rng seq c= dom f ; ::_thesis: ( f /* seq is convergent & lim (f /* seq) = lim_in+infty f1 ) consider k being Element of NAT such that A11: for n being Element of NAT st k <= n holds r1 < seq . n by A9, Def4; A12: seq ^\ k is divergent_to+infty by A9, Th26; now__::_thesis:_for_x_being_set_st_x_in_rng_(seq_^\_k)_holds_ x_in_right_open_halfline_r1 let x be set ; ::_thesis: ( x in rng (seq ^\ k) implies x in right_open_halfline r1 ) assume x in rng (seq ^\ k) ; ::_thesis: x in right_open_halfline r1 then consider n being Element of NAT such that A13: x = (seq ^\ k) . n by FUNCT_2:113; r1 < seq . (n + k) by A11, NAT_1:12; then r1 < (seq ^\ k) . n by NAT_1:def_3; then x in { g where g is Real : r1 < g } by A13; hence x in right_open_halfline r1 by XXREAL_1:230; ::_thesis: verum end; then A14: rng (seq ^\ k) c= right_open_halfline r1 by TARSKI:def_3; A15: rng (seq ^\ k) c= rng seq by VALUED_0:21; then rng (seq ^\ k) c= dom f by A10, XBOOLE_1:1; then A16: rng (seq ^\ k) c= (dom f) /\ (right_open_halfline r1) by A14, XBOOLE_1:19; then A17: rng (seq ^\ k) c= (dom f1) /\ (right_open_halfline r1) by A7, XBOOLE_1:1; then A18: rng (seq ^\ k) c= (dom f2) /\ (right_open_halfline r1) by A7, XBOOLE_1:1; A19: (dom f2) /\ (right_open_halfline r1) c= dom f2 by XBOOLE_1:17; then rng (seq ^\ k) c= dom f2 by A18, XBOOLE_1:1; then A20: ( f2 /* (seq ^\ k) is convergent & lim (f2 /* (seq ^\ k)) = lim_in+infty f1 ) by A2, A3, A12, Def12; A21: (dom f1) /\ (right_open_halfline r1) c= dom f1 by XBOOLE_1:17; then rng (seq ^\ k) c= dom f1 by A17, XBOOLE_1:1; then A22: ( f1 /* (seq ^\ k) is convergent & lim (f1 /* (seq ^\ k)) = lim_in+infty f1 ) by A1, A3, A12, Def12; A23: now__::_thesis:_for_n_being_Element_of_NAT_holds_ (_(f1_/*_(seq_^\_k))_._n_<=_(f_/*_(seq_^\_k))_._n_&_(f_/*_(seq_^\_k))_._n_<=_(f2_/*_(seq_^\_k))_._n_) let n be Element of NAT ; ::_thesis: ( (f1 /* (seq ^\ k)) . n <= (f /* (seq ^\ k)) . n & (f /* (seq ^\ k)) . n <= (f2 /* (seq ^\ k)) . n ) A24: (seq ^\ k) . n in rng (seq ^\ k) by VALUED_0:28; then f . ((seq ^\ k) . n) <= f2 . ((seq ^\ k) . n) by A6, A16; then A25: (f /* (seq ^\ k)) . n <= f2 . ((seq ^\ k) . n) by A10, A15, FUNCT_2:108, XBOOLE_1:1; f1 . ((seq ^\ k) . n) <= f . ((seq ^\ k) . n) by A6, A16, A24; then f1 . ((seq ^\ k) . n) <= (f /* (seq ^\ k)) . n by A10, A15, FUNCT_2:108, XBOOLE_1:1; hence ( (f1 /* (seq ^\ k)) . n <= (f /* (seq ^\ k)) . n & (f /* (seq ^\ k)) . n <= (f2 /* (seq ^\ k)) . n ) by A17, A21, A18, A19, A25, FUNCT_2:108, XBOOLE_1:1; ::_thesis: verum end; A26: f /* (seq ^\ k) = (f /* seq) ^\ k by A10, VALUED_0:27; then A27: (f /* seq) ^\ k is convergent by A22, A20, A23, SEQ_2:19; hence f /* seq is convergent by SEQ_4:21; ::_thesis: lim (f /* seq) = lim_in+infty f1 lim ((f /* seq) ^\ k) = lim_in+infty f1 by A22, A20, A23, A26, SEQ_2:20; hence lim (f /* seq) = lim_in+infty f1 by A27, SEQ_4:20, SEQ_4:21; ::_thesis: verum end; hence f is convergent_in+infty by A4, Def6; ::_thesis: ( f is convergent_in+infty & lim_in+infty f = lim_in+infty f1 ) hence ( f is convergent_in+infty & lim_in+infty f = lim_in+infty f1 ) by A8, Def12; ::_thesis: verum end; supposeA28: ( (dom f2) /\ (right_open_halfline r1) c= (dom f1) /\ (right_open_halfline r1) & (dom f) /\ (right_open_halfline r1) c= (dom f2) /\ (right_open_halfline r1) ) ; ::_thesis: ( f is convergent_in+infty & f is convergent_in+infty & lim_in+infty f = lim_in+infty f1 ) A29: now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_divergent_to+infty_&_rng_seq_c=_dom_f_holds_ (_f_/*_seq_is_convergent_&_lim_(f_/*_seq)_=_lim_in+infty_f1_) let seq be Real_Sequence; ::_thesis: ( seq is divergent_to+infty & rng seq c= dom f implies ( f /* seq is convergent & lim (f /* seq) = lim_in+infty f1 ) ) assume that A30: seq is divergent_to+infty and A31: rng seq c= dom f ; ::_thesis: ( f /* seq is convergent & lim (f /* seq) = lim_in+infty f1 ) consider k being Element of NAT such that A32: for n being Element of NAT st k <= n holds r1 < seq . n by A30, Def4; A33: seq ^\ k is divergent_to+infty by A30, Th26; now__::_thesis:_for_x_being_set_st_x_in_rng_(seq_^\_k)_holds_ x_in_right_open_halfline_r1 let x be set ; ::_thesis: ( x in rng (seq ^\ k) implies x in right_open_halfline r1 ) assume x in rng (seq ^\ k) ; ::_thesis: x in right_open_halfline r1 then consider n being Element of NAT such that A34: x = (seq ^\ k) . n by FUNCT_2:113; r1 < seq . (n + k) by A32, NAT_1:12; then r1 < (seq ^\ k) . n by NAT_1:def_3; then x in { g where g is Real : r1 < g } by A34; hence x in right_open_halfline r1 by XXREAL_1:230; ::_thesis: verum end; then A35: rng (seq ^\ k) c= right_open_halfline r1 by TARSKI:def_3; A36: rng (seq ^\ k) c= rng seq by VALUED_0:21; then rng (seq ^\ k) c= dom f by A31, XBOOLE_1:1; then A37: rng (seq ^\ k) c= (dom f) /\ (right_open_halfline r1) by A35, XBOOLE_1:19; then A38: rng (seq ^\ k) c= (dom f2) /\ (right_open_halfline r1) by A28, XBOOLE_1:1; then A39: rng (seq ^\ k) c= (dom f1) /\ (right_open_halfline r1) by A28, XBOOLE_1:1; A40: (dom f1) /\ (right_open_halfline r1) c= dom f1 by XBOOLE_1:17; then rng (seq ^\ k) c= dom f1 by A39, XBOOLE_1:1; then A41: ( f1 /* (seq ^\ k) is convergent & lim (f1 /* (seq ^\ k)) = lim_in+infty f1 ) by A1, A3, A33, Def12; A42: (dom f2) /\ (right_open_halfline r1) c= dom f2 by XBOOLE_1:17; then rng (seq ^\ k) c= dom f2 by A38, XBOOLE_1:1; then A43: ( f2 /* (seq ^\ k) is convergent & lim (f2 /* (seq ^\ k)) = lim_in+infty f1 ) by A2, A3, A33, Def12; A44: now__::_thesis:_for_n_being_Element_of_NAT_holds_ (_(f1_/*_(seq_^\_k))_._n_<=_(f_/*_(seq_^\_k))_._n_&_(f_/*_(seq_^\_k))_._n_<=_(f2_/*_(seq_^\_k))_._n_) let n be Element of NAT ; ::_thesis: ( (f1 /* (seq ^\ k)) . n <= (f /* (seq ^\ k)) . n & (f /* (seq ^\ k)) . n <= (f2 /* (seq ^\ k)) . n ) A45: (seq ^\ k) . n in rng (seq ^\ k) by VALUED_0:28; then f . ((seq ^\ k) . n) <= f2 . ((seq ^\ k) . n) by A6, A37; then A46: (f /* (seq ^\ k)) . n <= f2 . ((seq ^\ k) . n) by A31, A36, FUNCT_2:108, XBOOLE_1:1; f1 . ((seq ^\ k) . n) <= f . ((seq ^\ k) . n) by A6, A37, A45; then f1 . ((seq ^\ k) . n) <= (f /* (seq ^\ k)) . n by A31, A36, FUNCT_2:108, XBOOLE_1:1; hence ( (f1 /* (seq ^\ k)) . n <= (f /* (seq ^\ k)) . n & (f /* (seq ^\ k)) . n <= (f2 /* (seq ^\ k)) . n ) by A38, A42, A39, A40, A46, FUNCT_2:108, XBOOLE_1:1; ::_thesis: verum end; A47: f /* (seq ^\ k) = (f /* seq) ^\ k by A31, VALUED_0:27; then A48: (f /* seq) ^\ k is convergent by A43, A41, A44, SEQ_2:19; hence f /* seq is convergent by SEQ_4:21; ::_thesis: lim (f /* seq) = lim_in+infty f1 lim ((f /* seq) ^\ k) = lim_in+infty f1 by A43, A41, A44, A47, SEQ_2:20; hence lim (f /* seq) = lim_in+infty f1 by A48, SEQ_4:20, SEQ_4:21; ::_thesis: verum end; hence f is convergent_in+infty by A4, Def6; ::_thesis: ( f is convergent_in+infty & lim_in+infty f = lim_in+infty f1 ) hence ( f is convergent_in+infty & lim_in+infty f = lim_in+infty f1 ) by A29, Def12; ::_thesis: verum end; end; end; hence ( f is convergent_in+infty & lim_in+infty f = lim_in+infty f1 ) ; ::_thesis: verum end; theorem :: LIMFUNC1:101 for f1, f2, f being PartFunc of REAL,REAL st f1 is convergent_in+infty & f2 is convergent_in+infty & lim_in+infty f1 = lim_in+infty f2 & ex r being Real st ( right_open_halfline r c= ((dom f1) /\ (dom f2)) /\ (dom f) & ( for g being Real st g in right_open_halfline r holds ( f1 . g <= f . g & f . g <= f2 . g ) ) ) holds ( f is convergent_in+infty & lim_in+infty f = lim_in+infty f1 ) proof let f1, f2, f be PartFunc of REAL,REAL; ::_thesis: ( f1 is convergent_in+infty & f2 is convergent_in+infty & lim_in+infty f1 = lim_in+infty f2 & ex r being Real st ( right_open_halfline r c= ((dom f1) /\ (dom f2)) /\ (dom f) & ( for g being Real st g in right_open_halfline r holds ( f1 . g <= f . g & f . g <= f2 . g ) ) ) implies ( f is convergent_in+infty & lim_in+infty f = lim_in+infty f1 ) ) assume A1: ( f1 is convergent_in+infty & f2 is convergent_in+infty & lim_in+infty f1 = lim_in+infty f2 ) ; ::_thesis: ( for r being Real holds ( not right_open_halfline r c= ((dom f1) /\ (dom f2)) /\ (dom f) or ex g being Real st ( g in right_open_halfline r & not ( f1 . g <= f . g & f . g <= f2 . g ) ) ) or ( f is convergent_in+infty & lim_in+infty f = lim_in+infty f1 ) ) given r1 being Real such that A2: right_open_halfline r1 c= ((dom f1) /\ (dom f2)) /\ (dom f) and A3: for g being Real st g in right_open_halfline r1 holds ( f1 . g <= f . g & f . g <= f2 . g ) ; ::_thesis: ( f is convergent_in+infty & lim_in+infty f = lim_in+infty f1 ) ((dom f1) /\ (dom f2)) /\ (dom f) c= dom f by XBOOLE_1:17; then A4: right_open_halfline r1 c= dom f by A2, XBOOLE_1:1; A5: now__::_thesis:_for_r_being_Real_ex_g_being_Real_st_ (_r_<_g_&_g_in_dom_f_) let r be Real; ::_thesis: ex g being Real st ( r < g & g in dom f ) consider g being real number such that A6: (abs r) + (abs r1) < g by XREAL_1:1; reconsider g = g as Real by XREAL_0:def_1; take g = g; ::_thesis: ( r < g & g in dom f ) ( r <= abs r & 0 <= abs r1 ) by ABSVALUE:4, COMPLEX1:46; then r + 0 <= (abs r) + (abs r1) by XREAL_1:7; hence r < g by A6, XXREAL_0:2; ::_thesis: g in dom f ( r1 <= abs r1 & 0 <= abs r ) by ABSVALUE:4, COMPLEX1:46; then 0 + r1 <= (abs r) + (abs r1) by XREAL_1:7; then r1 < g by A6, XXREAL_0:2; then g in { g1 where g1 is Real : r1 < g1 } ; then g in right_open_halfline r1 by XXREAL_1:230; hence g in dom f by A4; ::_thesis: verum end; A7: ((dom f1) /\ (dom f2)) /\ (dom f) c= (dom f1) /\ (dom f2) by XBOOLE_1:17; now__::_thesis:_(_(dom_f1)_/\_(right_open_halfline_r1)_c=_(dom_f2)_/\_(right_open_halfline_r1)_&_(dom_f)_/\_(right_open_halfline_r1)_c=_(dom_f1)_/\_(right_open_halfline_r1)_&_(_for_g_being_Real_st_g_in_(dom_f)_/\_(right_open_halfline_r1)_holds_ (_f1_._g_<=_f_._g_&_f_._g_<=_f2_._g_)_)_) (dom f1) /\ (dom f2) c= dom f1 by XBOOLE_1:17; then ((dom f1) /\ (dom f2)) /\ (dom f) c= dom f1 by A7, XBOOLE_1:1; then A8: (dom f1) /\ (right_open_halfline r1) = right_open_halfline r1 by A2, XBOOLE_1:1, XBOOLE_1:28; (dom f1) /\ (dom f2) c= dom f2 by XBOOLE_1:17; then ((dom f1) /\ (dom f2)) /\ (dom f) c= dom f2 by A7, XBOOLE_1:1; hence (dom f1) /\ (right_open_halfline r1) c= (dom f2) /\ (right_open_halfline r1) by A2, A8, XBOOLE_1:1, XBOOLE_1:28; ::_thesis: ( (dom f) /\ (right_open_halfline r1) c= (dom f1) /\ (right_open_halfline r1) & ( for g being Real st g in (dom f) /\ (right_open_halfline r1) holds ( f1 . g <= f . g & f . g <= f2 . g ) ) ) thus (dom f) /\ (right_open_halfline r1) c= (dom f1) /\ (right_open_halfline r1) by A8, XBOOLE_1:17; ::_thesis: for g being Real st g in (dom f) /\ (right_open_halfline r1) holds ( f1 . g <= f . g & f . g <= f2 . g ) let g be Real; ::_thesis: ( g in (dom f) /\ (right_open_halfline r1) implies ( f1 . g <= f . g & f . g <= f2 . g ) ) assume g in (dom f) /\ (right_open_halfline r1) ; ::_thesis: ( f1 . g <= f . g & f . g <= f2 . g ) then g in right_open_halfline r1 by XBOOLE_0:def_4; hence ( f1 . g <= f . g & f . g <= f2 . g ) by A3; ::_thesis: verum end; hence ( f is convergent_in+infty & lim_in+infty f = lim_in+infty f1 ) by A1, A5, Th100; ::_thesis: verum end; theorem Th102: :: LIMFUNC1:102 for f1, f2, f being PartFunc of REAL,REAL st f1 is convergent_in-infty & f2 is convergent_in-infty & lim_in-infty f1 = lim_in-infty f2 & ( for r being Real ex g being Real st ( g < r & g in dom f ) ) & ex r being Real st ( ( ( (dom f1) /\ (left_open_halfline r) c= (dom f2) /\ (left_open_halfline r) & (dom f) /\ (left_open_halfline r) c= (dom f1) /\ (left_open_halfline r) ) or ( (dom f2) /\ (left_open_halfline r) c= (dom f1) /\ (left_open_halfline r) & (dom f) /\ (left_open_halfline r) c= (dom f2) /\ (left_open_halfline r) ) ) & ( for g being Real st g in (dom f) /\ (left_open_halfline r) holds ( f1 . g <= f . g & f . g <= f2 . g ) ) ) holds ( f is convergent_in-infty & lim_in-infty f = lim_in-infty f1 ) proof let f1, f2, f be PartFunc of REAL,REAL; ::_thesis: ( f1 is convergent_in-infty & f2 is convergent_in-infty & lim_in-infty f1 = lim_in-infty f2 & ( for r being Real ex g being Real st ( g < r & g in dom f ) ) & ex r being Real st ( ( ( (dom f1) /\ (left_open_halfline r) c= (dom f2) /\ (left_open_halfline r) & (dom f) /\ (left_open_halfline r) c= (dom f1) /\ (left_open_halfline r) ) or ( (dom f2) /\ (left_open_halfline r) c= (dom f1) /\ (left_open_halfline r) & (dom f) /\ (left_open_halfline r) c= (dom f2) /\ (left_open_halfline r) ) ) & ( for g being Real st g in (dom f) /\ (left_open_halfline r) holds ( f1 . g <= f . g & f . g <= f2 . g ) ) ) implies ( f is convergent_in-infty & lim_in-infty f = lim_in-infty f1 ) ) assume that A1: f1 is convergent_in-infty and A2: f2 is convergent_in-infty and A3: lim_in-infty f1 = lim_in-infty f2 and A4: for r being Real ex g being Real st ( g < r & g in dom f ) ; ::_thesis: ( for r being Real holds ( ( not ( (dom f1) /\ (left_open_halfline r) c= (dom f2) /\ (left_open_halfline r) & (dom f) /\ (left_open_halfline r) c= (dom f1) /\ (left_open_halfline r) ) & not ( (dom f2) /\ (left_open_halfline r) c= (dom f1) /\ (left_open_halfline r) & (dom f) /\ (left_open_halfline r) c= (dom f2) /\ (left_open_halfline r) ) ) or ex g being Real st ( g in (dom f) /\ (left_open_halfline r) & not ( f1 . g <= f . g & f . g <= f2 . g ) ) ) or ( f is convergent_in-infty & lim_in-infty f = lim_in-infty f1 ) ) given r1 being Real such that A5: ( ( (dom f1) /\ (left_open_halfline r1) c= (dom f2) /\ (left_open_halfline r1) & (dom f) /\ (left_open_halfline r1) c= (dom f1) /\ (left_open_halfline r1) ) or ( (dom f2) /\ (left_open_halfline r1) c= (dom f1) /\ (left_open_halfline r1) & (dom f) /\ (left_open_halfline r1) c= (dom f2) /\ (left_open_halfline r1) ) ) and A6: for g being Real st g in (dom f) /\ (left_open_halfline r1) holds ( f1 . g <= f . g & f . g <= f2 . g ) ; ::_thesis: ( f is convergent_in-infty & lim_in-infty f = lim_in-infty f1 ) now__::_thesis:_(_f_is_convergent_in-infty_&_f_is_convergent_in-infty_&_lim_in-infty_f_=_lim_in-infty_f1_) percases ( ( (dom f1) /\ (left_open_halfline r1) c= (dom f2) /\ (left_open_halfline r1) & (dom f) /\ (left_open_halfline r1) c= (dom f1) /\ (left_open_halfline r1) ) or ( (dom f2) /\ (left_open_halfline r1) c= (dom f1) /\ (left_open_halfline r1) & (dom f) /\ (left_open_halfline r1) c= (dom f2) /\ (left_open_halfline r1) ) ) by A5; supposeA7: ( (dom f1) /\ (left_open_halfline r1) c= (dom f2) /\ (left_open_halfline r1) & (dom f) /\ (left_open_halfline r1) c= (dom f1) /\ (left_open_halfline r1) ) ; ::_thesis: ( f is convergent_in-infty & f is convergent_in-infty & lim_in-infty f = lim_in-infty f1 ) A8: now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_divergent_to-infty_&_rng_seq_c=_dom_f_holds_ (_f_/*_seq_is_convergent_&_lim_(f_/*_seq)_=_lim_in-infty_f1_) let seq be Real_Sequence; ::_thesis: ( seq is divergent_to-infty & rng seq c= dom f implies ( f /* seq is convergent & lim (f /* seq) = lim_in-infty f1 ) ) assume that A9: seq is divergent_to-infty and A10: rng seq c= dom f ; ::_thesis: ( f /* seq is convergent & lim (f /* seq) = lim_in-infty f1 ) consider k being Element of NAT such that A11: for n being Element of NAT st k <= n holds seq . n < r1 by A9, Def5; A12: seq ^\ k is divergent_to-infty by A9, Th27; now__::_thesis:_for_x_being_set_st_x_in_rng_(seq_^\_k)_holds_ x_in_left_open_halfline_r1 let x be set ; ::_thesis: ( x in rng (seq ^\ k) implies x in left_open_halfline r1 ) assume x in rng (seq ^\ k) ; ::_thesis: x in left_open_halfline r1 then consider n being Element of NAT such that A13: x = (seq ^\ k) . n by FUNCT_2:113; seq . (n + k) < r1 by A11, NAT_1:12; then (seq ^\ k) . n < r1 by NAT_1:def_3; then x in { g where g is Real : g < r1 } by A13; hence x in left_open_halfline r1 by XXREAL_1:229; ::_thesis: verum end; then A14: rng (seq ^\ k) c= left_open_halfline r1 by TARSKI:def_3; A15: rng (seq ^\ k) c= rng seq by VALUED_0:21; then rng (seq ^\ k) c= dom f by A10, XBOOLE_1:1; then A16: rng (seq ^\ k) c= (dom f) /\ (left_open_halfline r1) by A14, XBOOLE_1:19; then A17: rng (seq ^\ k) c= (dom f1) /\ (left_open_halfline r1) by A7, XBOOLE_1:1; then A18: rng (seq ^\ k) c= (dom f2) /\ (left_open_halfline r1) by A7, XBOOLE_1:1; A19: (dom f2) /\ (left_open_halfline r1) c= dom f2 by XBOOLE_1:17; then rng (seq ^\ k) c= dom f2 by A18, XBOOLE_1:1; then A20: ( f2 /* (seq ^\ k) is convergent & lim (f2 /* (seq ^\ k)) = lim_in-infty f1 ) by A2, A3, A12, Def13; A21: (dom f1) /\ (left_open_halfline r1) c= dom f1 by XBOOLE_1:17; then rng (seq ^\ k) c= dom f1 by A17, XBOOLE_1:1; then A22: ( f1 /* (seq ^\ k) is convergent & lim (f1 /* (seq ^\ k)) = lim_in-infty f1 ) by A1, A3, A12, Def13; A23: now__::_thesis:_for_n_being_Element_of_NAT_holds_ (_(f1_/*_(seq_^\_k))_._n_<=_(f_/*_(seq_^\_k))_._n_&_(f_/*_(seq_^\_k))_._n_<=_(f2_/*_(seq_^\_k))_._n_) let n be Element of NAT ; ::_thesis: ( (f1 /* (seq ^\ k)) . n <= (f /* (seq ^\ k)) . n & (f /* (seq ^\ k)) . n <= (f2 /* (seq ^\ k)) . n ) A24: (seq ^\ k) . n in rng (seq ^\ k) by VALUED_0:28; then f . ((seq ^\ k) . n) <= f2 . ((seq ^\ k) . n) by A6, A16; then A25: (f /* (seq ^\ k)) . n <= f2 . ((seq ^\ k) . n) by A10, A15, FUNCT_2:108, XBOOLE_1:1; f1 . ((seq ^\ k) . n) <= f . ((seq ^\ k) . n) by A6, A16, A24; then f1 . ((seq ^\ k) . n) <= (f /* (seq ^\ k)) . n by A10, A15, FUNCT_2:108, XBOOLE_1:1; hence ( (f1 /* (seq ^\ k)) . n <= (f /* (seq ^\ k)) . n & (f /* (seq ^\ k)) . n <= (f2 /* (seq ^\ k)) . n ) by A17, A21, A18, A19, A25, FUNCT_2:108, XBOOLE_1:1; ::_thesis: verum end; A26: f /* (seq ^\ k) = (f /* seq) ^\ k by A10, VALUED_0:27; then A27: (f /* seq) ^\ k is convergent by A22, A20, A23, SEQ_2:19; hence f /* seq is convergent by SEQ_4:21; ::_thesis: lim (f /* seq) = lim_in-infty f1 lim ((f /* seq) ^\ k) = lim_in-infty f1 by A22, A20, A23, A26, SEQ_2:20; hence lim (f /* seq) = lim_in-infty f1 by A27, SEQ_4:20, SEQ_4:21; ::_thesis: verum end; hence f is convergent_in-infty by A4, Def9; ::_thesis: ( f is convergent_in-infty & lim_in-infty f = lim_in-infty f1 ) hence ( f is convergent_in-infty & lim_in-infty f = lim_in-infty f1 ) by A8, Def13; ::_thesis: verum end; supposeA28: ( (dom f2) /\ (left_open_halfline r1) c= (dom f1) /\ (left_open_halfline r1) & (dom f) /\ (left_open_halfline r1) c= (dom f2) /\ (left_open_halfline r1) ) ; ::_thesis: ( f is convergent_in-infty & f is convergent_in-infty & lim_in-infty f = lim_in-infty f1 ) A29: now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_divergent_to-infty_&_rng_seq_c=_dom_f_holds_ (_f_/*_seq_is_convergent_&_lim_(f_/*_seq)_=_lim_in-infty_f1_) let seq be Real_Sequence; ::_thesis: ( seq is divergent_to-infty & rng seq c= dom f implies ( f /* seq is convergent & lim (f /* seq) = lim_in-infty f1 ) ) assume that A30: seq is divergent_to-infty and A31: rng seq c= dom f ; ::_thesis: ( f /* seq is convergent & lim (f /* seq) = lim_in-infty f1 ) consider k being Element of NAT such that A32: for n being Element of NAT st k <= n holds seq . n < r1 by A30, Def5; A33: seq ^\ k is divergent_to-infty by A30, Th27; now__::_thesis:_for_x_being_set_st_x_in_rng_(seq_^\_k)_holds_ x_in_left_open_halfline_r1 let x be set ; ::_thesis: ( x in rng (seq ^\ k) implies x in left_open_halfline r1 ) assume x in rng (seq ^\ k) ; ::_thesis: x in left_open_halfline r1 then consider n being Element of NAT such that A34: x = (seq ^\ k) . n by FUNCT_2:113; seq . (n + k) < r1 by A32, NAT_1:12; then (seq ^\ k) . n < r1 by NAT_1:def_3; then x in { g where g is Real : g < r1 } by A34; hence x in left_open_halfline r1 by XXREAL_1:229; ::_thesis: verum end; then A35: rng (seq ^\ k) c= left_open_halfline r1 by TARSKI:def_3; A36: rng (seq ^\ k) c= rng seq by VALUED_0:21; then rng (seq ^\ k) c= dom f by A31, XBOOLE_1:1; then A37: rng (seq ^\ k) c= (dom f) /\ (left_open_halfline r1) by A35, XBOOLE_1:19; then A38: rng (seq ^\ k) c= (dom f2) /\ (left_open_halfline r1) by A28, XBOOLE_1:1; then A39: rng (seq ^\ k) c= (dom f1) /\ (left_open_halfline r1) by A28, XBOOLE_1:1; A40: (dom f1) /\ (left_open_halfline r1) c= dom f1 by XBOOLE_1:17; then rng (seq ^\ k) c= dom f1 by A39, XBOOLE_1:1; then A41: ( f1 /* (seq ^\ k) is convergent & lim (f1 /* (seq ^\ k)) = lim_in-infty f1 ) by A1, A3, A33, Def13; A42: (dom f2) /\ (left_open_halfline r1) c= dom f2 by XBOOLE_1:17; then rng (seq ^\ k) c= dom f2 by A38, XBOOLE_1:1; then A43: ( f2 /* (seq ^\ k) is convergent & lim (f2 /* (seq ^\ k)) = lim_in-infty f1 ) by A2, A3, A33, Def13; A44: now__::_thesis:_for_n_being_Element_of_NAT_holds_ (_(f1_/*_(seq_^\_k))_._n_<=_(f_/*_(seq_^\_k))_._n_&_(f_/*_(seq_^\_k))_._n_<=_(f2_/*_(seq_^\_k))_._n_) let n be Element of NAT ; ::_thesis: ( (f1 /* (seq ^\ k)) . n <= (f /* (seq ^\ k)) . n & (f /* (seq ^\ k)) . n <= (f2 /* (seq ^\ k)) . n ) A45: (seq ^\ k) . n in rng (seq ^\ k) by VALUED_0:28; then f . ((seq ^\ k) . n) <= f2 . ((seq ^\ k) . n) by A6, A37; then A46: (f /* (seq ^\ k)) . n <= f2 . ((seq ^\ k) . n) by A31, A36, FUNCT_2:108, XBOOLE_1:1; f1 . ((seq ^\ k) . n) <= f . ((seq ^\ k) . n) by A6, A37, A45; then f1 . ((seq ^\ k) . n) <= (f /* (seq ^\ k)) . n by A31, A36, FUNCT_2:108, XBOOLE_1:1; hence ( (f1 /* (seq ^\ k)) . n <= (f /* (seq ^\ k)) . n & (f /* (seq ^\ k)) . n <= (f2 /* (seq ^\ k)) . n ) by A38, A42, A39, A40, A46, FUNCT_2:108, XBOOLE_1:1; ::_thesis: verum end; A47: f /* (seq ^\ k) = (f /* seq) ^\ k by A31, VALUED_0:27; then A48: (f /* seq) ^\ k is convergent by A43, A41, A44, SEQ_2:19; hence f /* seq is convergent by SEQ_4:21; ::_thesis: lim (f /* seq) = lim_in-infty f1 lim ((f /* seq) ^\ k) = lim_in-infty f1 by A43, A41, A44, A47, SEQ_2:20; hence lim (f /* seq) = lim_in-infty f1 by A48, SEQ_4:20, SEQ_4:21; ::_thesis: verum end; hence f is convergent_in-infty by A4, Def9; ::_thesis: ( f is convergent_in-infty & lim_in-infty f = lim_in-infty f1 ) hence ( f is convergent_in-infty & lim_in-infty f = lim_in-infty f1 ) by A29, Def13; ::_thesis: verum end; end; end; hence ( f is convergent_in-infty & lim_in-infty f = lim_in-infty f1 ) ; ::_thesis: verum end; theorem :: LIMFUNC1:103 for f1, f2, f being PartFunc of REAL,REAL st f1 is convergent_in-infty & f2 is convergent_in-infty & lim_in-infty f1 = lim_in-infty f2 & ex r being Real st ( left_open_halfline r c= ((dom f1) /\ (dom f2)) /\ (dom f) & ( for g being Real st g in left_open_halfline r holds ( f1 . g <= f . g & f . g <= f2 . g ) ) ) holds ( f is convergent_in-infty & lim_in-infty f = lim_in-infty f1 ) proof let f1, f2, f be PartFunc of REAL,REAL; ::_thesis: ( f1 is convergent_in-infty & f2 is convergent_in-infty & lim_in-infty f1 = lim_in-infty f2 & ex r being Real st ( left_open_halfline r c= ((dom f1) /\ (dom f2)) /\ (dom f) & ( for g being Real st g in left_open_halfline r holds ( f1 . g <= f . g & f . g <= f2 . g ) ) ) implies ( f is convergent_in-infty & lim_in-infty f = lim_in-infty f1 ) ) assume A1: ( f1 is convergent_in-infty & f2 is convergent_in-infty & lim_in-infty f1 = lim_in-infty f2 ) ; ::_thesis: ( for r being Real holds ( not left_open_halfline r c= ((dom f1) /\ (dom f2)) /\ (dom f) or ex g being Real st ( g in left_open_halfline r & not ( f1 . g <= f . g & f . g <= f2 . g ) ) ) or ( f is convergent_in-infty & lim_in-infty f = lim_in-infty f1 ) ) given r1 being Real such that A2: left_open_halfline r1 c= ((dom f1) /\ (dom f2)) /\ (dom f) and A3: for g being Real st g in left_open_halfline r1 holds ( f1 . g <= f . g & f . g <= f2 . g ) ; ::_thesis: ( f is convergent_in-infty & lim_in-infty f = lim_in-infty f1 ) ((dom f1) /\ (dom f2)) /\ (dom f) c= dom f by XBOOLE_1:17; then A4: left_open_halfline r1 c= dom f by A2, XBOOLE_1:1; A5: now__::_thesis:_for_r_being_Real_ex_g_being_Real_st_ (_g_<_r_&_g_in_dom_f_) let r be Real; ::_thesis: ex g being Real st ( g < r & g in dom f ) consider g being real number such that A6: g < (- (abs r)) - (abs r1) by XREAL_1:2; reconsider g = g as Real by XREAL_0:def_1; take g = g; ::_thesis: ( g < r & g in dom f ) ( - (abs r) <= r & 0 <= abs r1 ) by ABSVALUE:4, COMPLEX1:46; then (- (abs r)) - (abs r1) <= r - 0 by XREAL_1:13; hence g < r by A6, XXREAL_0:2; ::_thesis: g in dom f ( - (abs r1) <= r1 & 0 <= abs r ) by ABSVALUE:4, COMPLEX1:46; then (- (abs r1)) - (abs r) <= r1 - 0 by XREAL_1:13; then g < r1 by A6, XXREAL_0:2; then g in { g1 where g1 is Real : g1 < r1 } ; then g in left_open_halfline r1 by XXREAL_1:229; hence g in dom f by A4; ::_thesis: verum end; A7: ((dom f1) /\ (dom f2)) /\ (dom f) c= (dom f1) /\ (dom f2) by XBOOLE_1:17; now__::_thesis:_(_(dom_f1)_/\_(left_open_halfline_r1)_c=_(dom_f2)_/\_(left_open_halfline_r1)_&_(dom_f)_/\_(left_open_halfline_r1)_c=_(dom_f1)_/\_(left_open_halfline_r1)_&_(_for_g_being_Real_st_g_in_(dom_f)_/\_(left_open_halfline_r1)_holds_ (_f1_._g_<=_f_._g_&_f_._g_<=_f2_._g_)_)_) (dom f1) /\ (dom f2) c= dom f1 by XBOOLE_1:17; then ((dom f1) /\ (dom f2)) /\ (dom f) c= dom f1 by A7, XBOOLE_1:1; then A8: (dom f1) /\ (left_open_halfline r1) = left_open_halfline r1 by A2, XBOOLE_1:1, XBOOLE_1:28; (dom f1) /\ (dom f2) c= dom f2 by XBOOLE_1:17; then ((dom f1) /\ (dom f2)) /\ (dom f) c= dom f2 by A7, XBOOLE_1:1; hence (dom f1) /\ (left_open_halfline r1) c= (dom f2) /\ (left_open_halfline r1) by A2, A8, XBOOLE_1:1, XBOOLE_1:28; ::_thesis: ( (dom f) /\ (left_open_halfline r1) c= (dom f1) /\ (left_open_halfline r1) & ( for g being Real st g in (dom f) /\ (left_open_halfline r1) holds ( f1 . g <= f . g & f . g <= f2 . g ) ) ) thus (dom f) /\ (left_open_halfline r1) c= (dom f1) /\ (left_open_halfline r1) by A8, XBOOLE_1:17; ::_thesis: for g being Real st g in (dom f) /\ (left_open_halfline r1) holds ( f1 . g <= f . g & f . g <= f2 . g ) let g be Real; ::_thesis: ( g in (dom f) /\ (left_open_halfline r1) implies ( f1 . g <= f . g & f . g <= f2 . g ) ) assume g in (dom f) /\ (left_open_halfline r1) ; ::_thesis: ( f1 . g <= f . g & f . g <= f2 . g ) then g in left_open_halfline r1 by XBOOLE_0:def_4; hence ( f1 . g <= f . g & f . g <= f2 . g ) by A3; ::_thesis: verum end; hence ( f is convergent_in-infty & lim_in-infty f = lim_in-infty f1 ) by A1, A5, Th102; ::_thesis: verum end; theorem :: LIMFUNC1:104 for f1, f2 being PartFunc of REAL,REAL st f1 is convergent_in+infty & f2 is convergent_in+infty & ex r being Real st ( ( (dom f1) /\ (right_open_halfline r) c= (dom f2) /\ (right_open_halfline r) & ( for g being Real st g in (dom f1) /\ (right_open_halfline r) holds f1 . g <= f2 . g ) ) or ( (dom f2) /\ (right_open_halfline r) c= (dom f1) /\ (right_open_halfline r) & ( for g being Real st g in (dom f2) /\ (right_open_halfline r) holds f1 . g <= f2 . g ) ) ) holds lim_in+infty f1 <= lim_in+infty f2 proof let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is convergent_in+infty & f2 is convergent_in+infty & ex r being Real st ( ( (dom f1) /\ (right_open_halfline r) c= (dom f2) /\ (right_open_halfline r) & ( for g being Real st g in (dom f1) /\ (right_open_halfline r) holds f1 . g <= f2 . g ) ) or ( (dom f2) /\ (right_open_halfline r) c= (dom f1) /\ (right_open_halfline r) & ( for g being Real st g in (dom f2) /\ (right_open_halfline r) holds f1 . g <= f2 . g ) ) ) implies lim_in+infty f1 <= lim_in+infty f2 ) assume that A1: f1 is convergent_in+infty and A2: f2 is convergent_in+infty ; ::_thesis: ( for r being Real holds ( not ( (dom f1) /\ (right_open_halfline r) c= (dom f2) /\ (right_open_halfline r) & ( for g being Real st g in (dom f1) /\ (right_open_halfline r) holds f1 . g <= f2 . g ) ) & not ( (dom f2) /\ (right_open_halfline r) c= (dom f1) /\ (right_open_halfline r) & ( for g being Real st g in (dom f2) /\ (right_open_halfline r) holds f1 . g <= f2 . g ) ) ) or lim_in+infty f1 <= lim_in+infty f2 ) given r being Real such that A3: ( ( (dom f1) /\ (right_open_halfline r) c= (dom f2) /\ (right_open_halfline r) & ( for g being Real st g in (dom f1) /\ (right_open_halfline r) holds f1 . g <= f2 . g ) ) or ( (dom f2) /\ (right_open_halfline r) c= (dom f1) /\ (right_open_halfline r) & ( for g being Real st g in (dom f2) /\ (right_open_halfline r) holds f1 . g <= f2 . g ) ) ) ; ::_thesis: lim_in+infty f1 <= lim_in+infty f2 now__::_thesis:_lim_in+infty_f1_<=_lim_in+infty_f2 percases ( ( (dom f1) /\ (right_open_halfline r) c= (dom f2) /\ (right_open_halfline r) & ( for g being Real st g in (dom f1) /\ (right_open_halfline r) holds f1 . g <= f2 . g ) ) or ( (dom f2) /\ (right_open_halfline r) c= (dom f1) /\ (right_open_halfline r) & ( for g being Real st g in (dom f2) /\ (right_open_halfline r) holds f1 . g <= f2 . g ) ) ) by A3; supposeA4: ( (dom f1) /\ (right_open_halfline r) c= (dom f2) /\ (right_open_halfline r) & ( for g being Real st g in (dom f1) /\ (right_open_halfline r) holds f1 . g <= f2 . g ) ) ; ::_thesis: lim_in+infty f1 <= lim_in+infty f2 defpred S1[ Element of NAT , real number ] means ( $1 < $2 & $2 in (dom f1) /\ (right_open_halfline r) ); A5: now__::_thesis:_for_n_being_Element_of_NAT_ex_g_being_Real_st_S1[n,g] let n be Element of NAT ; ::_thesis: ex g being Real st S1[n,g] 0 <= abs r by COMPLEX1:46; then A6: n + 0 <= n + (abs r) by XREAL_1:7; consider g being Real such that A7: n + (abs r) < g and A8: g in dom f1 by A1, Def6; take g = g; ::_thesis: S1[n,g] ( 0 <= n & r <= abs r ) by ABSVALUE:4, NAT_1:2; then 0 + r <= n + (abs r) by XREAL_1:7; then r < g by A7, XXREAL_0:2; then g in { g2 where g2 is Real : r < g2 } ; then g in right_open_halfline r by XXREAL_1:230; hence S1[n,g] by A7, A8, A6, XBOOLE_0:def_4, XXREAL_0:2; ::_thesis: verum end; consider s2 being Real_Sequence such that A9: for n being Element of NAT holds S1[n,s2 . n] from FUNCT_2:sch_3(A5); now__::_thesis:_for_n_being_Element_of_NAT_holds_(incl_NAT)_._n_<=_s2_._n let n be Element of NAT ; ::_thesis: (incl NAT) . n <= s2 . n n < s2 . n by A9; hence (incl NAT) . n <= s2 . n by FUNCT_1:18; ::_thesis: verum end; then A10: s2 is divergent_to+infty by Lm4, Th20, Th42; A11: rng s2 c= dom f2 proof let x be real number ; :: according to MEMBERED:def_9 ::_thesis: ( not x in rng s2 or x in dom f2 ) assume x in rng s2 ; ::_thesis: x in dom f2 then ex n being Element of NAT st x = s2 . n by FUNCT_2:113; then x in (dom f1) /\ (right_open_halfline r) by A9; hence x in dom f2 by A4, XBOOLE_0:def_4; ::_thesis: verum end; then A12: lim (f2 /* s2) = lim_in+infty f2 by A2, A10, Def12; A13: rng s2 c= dom f1 proof let x be real number ; :: according to MEMBERED:def_9 ::_thesis: ( not x in rng s2 or x in dom f1 ) assume x in rng s2 ; ::_thesis: x in dom f1 then ex n being Element of NAT st x = s2 . n by FUNCT_2:113; then x in (dom f1) /\ (right_open_halfline r) by A9; hence x in dom f1 by XBOOLE_0:def_4; ::_thesis: verum end; A14: now__::_thesis:_for_n_being_Element_of_NAT_holds_(f1_/*_s2)_._n_<=_(f2_/*_s2)_._n let n be Element of NAT ; ::_thesis: (f1 /* s2) . n <= (f2 /* s2) . n f1 . (s2 . n) <= f2 . (s2 . n) by A4, A9; then (f1 /* s2) . n <= f2 . (s2 . n) by A13, FUNCT_2:108; hence (f1 /* s2) . n <= (f2 /* s2) . n by A11, FUNCT_2:108; ::_thesis: verum end; lim_in+infty f2 = lim_in+infty f2 ; then A15: f2 /* s2 is convergent by A2, A10, A11, Def12; lim_in+infty f1 = lim_in+infty f1 ; then A16: f1 /* s2 is convergent by A1, A10, A13, Def12; lim (f1 /* s2) = lim_in+infty f1 by A1, A10, A13, Def12; hence lim_in+infty f1 <= lim_in+infty f2 by A16, A15, A12, A14, SEQ_2:18; ::_thesis: verum end; supposeA17: ( (dom f2) /\ (right_open_halfline r) c= (dom f1) /\ (right_open_halfline r) & ( for g being Real st g in (dom f2) /\ (right_open_halfline r) holds f1 . g <= f2 . g ) ) ; ::_thesis: lim_in+infty f1 <= lim_in+infty f2 defpred S1[ Element of NAT , real number ] means ( $1 < $2 & $2 in (dom f2) /\ (right_open_halfline r) ); A18: now__::_thesis:_for_n_being_Element_of_NAT_ex_g_being_Real_st_S1[n,g] let n be Element of NAT ; ::_thesis: ex g being Real st S1[n,g] 0 <= abs r by COMPLEX1:46; then A19: n + 0 <= n + (abs r) by XREAL_1:7; consider g being Real such that A20: n + (abs r) < g and A21: g in dom f2 by A2, Def6; take g = g; ::_thesis: S1[n,g] ( 0 <= n & r <= abs r ) by ABSVALUE:4, NAT_1:2; then 0 + r <= n + (abs r) by XREAL_1:7; then r < g by A20, XXREAL_0:2; then g in { g2 where g2 is Real : r < g2 } ; then g in right_open_halfline r by XXREAL_1:230; hence S1[n,g] by A20, A21, A19, XBOOLE_0:def_4, XXREAL_0:2; ::_thesis: verum end; consider s2 being Real_Sequence such that A22: for n being Element of NAT holds S1[n,s2 . n] from FUNCT_2:sch_3(A18); now__::_thesis:_for_n_being_Element_of_NAT_holds_(incl_NAT)_._n_<=_s2_._n let n be Element of NAT ; ::_thesis: (incl NAT) . n <= s2 . n n < s2 . n by A22; hence (incl NAT) . n <= s2 . n by FUNCT_1:18; ::_thesis: verum end; then A23: s2 is divergent_to+infty by Lm4, Th20, Th42; A24: rng s2 c= dom f1 proof let x be real number ; :: according to MEMBERED:def_9 ::_thesis: ( not x in rng s2 or x in dom f1 ) assume x in rng s2 ; ::_thesis: x in dom f1 then ex n being Element of NAT st x = s2 . n by FUNCT_2:113; then x in (dom f2) /\ (right_open_halfline r) by A22; hence x in dom f1 by A17, XBOOLE_0:def_4; ::_thesis: verum end; then A25: lim (f1 /* s2) = lim_in+infty f1 by A1, A23, Def12; A26: rng s2 c= dom f2 proof let x be real number ; :: according to MEMBERED:def_9 ::_thesis: ( not x in rng s2 or x in dom f2 ) assume x in rng s2 ; ::_thesis: x in dom f2 then ex n being Element of NAT st x = s2 . n by FUNCT_2:113; then x in (dom f2) /\ (right_open_halfline r) by A22; hence x in dom f2 by XBOOLE_0:def_4; ::_thesis: verum end; A27: now__::_thesis:_for_n_being_Element_of_NAT_holds_(f1_/*_s2)_._n_<=_(f2_/*_s2)_._n let n be Element of NAT ; ::_thesis: (f1 /* s2) . n <= (f2 /* s2) . n f1 . (s2 . n) <= f2 . (s2 . n) by A17, A22; then (f1 /* s2) . n <= f2 . (s2 . n) by A24, FUNCT_2:108; hence (f1 /* s2) . n <= (f2 /* s2) . n by A26, FUNCT_2:108; ::_thesis: verum end; lim_in+infty f1 = lim_in+infty f1 ; then A28: f1 /* s2 is convergent by A1, A23, A24, Def12; lim_in+infty f2 = lim_in+infty f2 ; then A29: f2 /* s2 is convergent by A2, A23, A26, Def12; lim (f2 /* s2) = lim_in+infty f2 by A2, A23, A26, Def12; hence lim_in+infty f1 <= lim_in+infty f2 by A29, A28, A25, A27, SEQ_2:18; ::_thesis: verum end; end; end; hence lim_in+infty f1 <= lim_in+infty f2 ; ::_thesis: verum end; theorem :: LIMFUNC1:105 for f1, f2 being PartFunc of REAL,REAL st f1 is convergent_in-infty & f2 is convergent_in-infty & ex r being Real st ( ( (dom f1) /\ (left_open_halfline r) c= (dom f2) /\ (left_open_halfline r) & ( for g being Real st g in (dom f1) /\ (left_open_halfline r) holds f1 . g <= f2 . g ) ) or ( (dom f2) /\ (left_open_halfline r) c= (dom f1) /\ (left_open_halfline r) & ( for g being Real st g in (dom f2) /\ (left_open_halfline r) holds f1 . g <= f2 . g ) ) ) holds lim_in-infty f1 <= lim_in-infty f2 proof let f1, f2 be PartFunc of REAL,REAL; ::_thesis: ( f1 is convergent_in-infty & f2 is convergent_in-infty & ex r being Real st ( ( (dom f1) /\ (left_open_halfline r) c= (dom f2) /\ (left_open_halfline r) & ( for g being Real st g in (dom f1) /\ (left_open_halfline r) holds f1 . g <= f2 . g ) ) or ( (dom f2) /\ (left_open_halfline r) c= (dom f1) /\ (left_open_halfline r) & ( for g being Real st g in (dom f2) /\ (left_open_halfline r) holds f1 . g <= f2 . g ) ) ) implies lim_in-infty f1 <= lim_in-infty f2 ) assume that A1: f1 is convergent_in-infty and A2: f2 is convergent_in-infty ; ::_thesis: ( for r being Real holds ( not ( (dom f1) /\ (left_open_halfline r) c= (dom f2) /\ (left_open_halfline r) & ( for g being Real st g in (dom f1) /\ (left_open_halfline r) holds f1 . g <= f2 . g ) ) & not ( (dom f2) /\ (left_open_halfline r) c= (dom f1) /\ (left_open_halfline r) & ( for g being Real st g in (dom f2) /\ (left_open_halfline r) holds f1 . g <= f2 . g ) ) ) or lim_in-infty f1 <= lim_in-infty f2 ) given r being Real such that A3: ( ( (dom f1) /\ (left_open_halfline r) c= (dom f2) /\ (left_open_halfline r) & ( for g being Real st g in (dom f1) /\ (left_open_halfline r) holds f1 . g <= f2 . g ) ) or ( (dom f2) /\ (left_open_halfline r) c= (dom f1) /\ (left_open_halfline r) & ( for g being Real st g in (dom f2) /\ (left_open_halfline r) holds f1 . g <= f2 . g ) ) ) ; ::_thesis: lim_in-infty f1 <= lim_in-infty f2 now__::_thesis:_lim_in-infty_f1_<=_lim_in-infty_f2 percases ( ( (dom f1) /\ (left_open_halfline r) c= (dom f2) /\ (left_open_halfline r) & ( for g being Real st g in (dom f1) /\ (left_open_halfline r) holds f1 . g <= f2 . g ) ) or ( (dom f2) /\ (left_open_halfline r) c= (dom f1) /\ (left_open_halfline r) & ( for g being Real st g in (dom f2) /\ (left_open_halfline r) holds f1 . g <= f2 . g ) ) ) by A3; supposeA4: ( (dom f1) /\ (left_open_halfline r) c= (dom f2) /\ (left_open_halfline r) & ( for g being Real st g in (dom f1) /\ (left_open_halfline r) holds f1 . g <= f2 . g ) ) ; ::_thesis: lim_in-infty f1 <= lim_in-infty f2 deffunc H1( Element of NAT ) -> Element of REAL = - $1; defpred S1[ Element of NAT , real number ] means ( $2 < - $1 & $2 in (dom f1) /\ (left_open_halfline r) ); consider s1 being Real_Sequence such that A5: for n being Element of NAT holds s1 . n = H1(n) from SEQ_1:sch_1(); A6: now__::_thesis:_for_n_being_Element_of_NAT_ex_g_being_Real_st_S1[n,g] let n be Element of NAT ; ::_thesis: ex g being Real st S1[n,g] 0 <= abs r by COMPLEX1:46; then A7: (- n) - (abs r) <= (- n) - 0 by XREAL_1:13; consider g being Real such that A8: g < (- n) - (abs r) and A9: g in dom f1 by A1, Def9; take g = g; ::_thesis: S1[n,g] ( 0 <= n & - (abs r) <= r ) by ABSVALUE:4, NAT_1:2; then (- (abs r)) - n <= r - 0 by XREAL_1:13; then g < r by A8, XXREAL_0:2; then g in { g2 where g2 is Real : g2 < r } ; then g in left_open_halfline r by XXREAL_1:229; hence S1[n,g] by A8, A9, A7, XBOOLE_0:def_4, XXREAL_0:2; ::_thesis: verum end; consider s2 being Real_Sequence such that A10: for n being Element of NAT holds S1[n,s2 . n] from FUNCT_2:sch_3(A6); now__::_thesis:_for_n_being_Element_of_NAT_holds_s2_._n_<=_s1_._n let n be Element of NAT ; ::_thesis: s2 . n <= s1 . n s2 . n < - n by A10; hence s2 . n <= s1 . n by A5; ::_thesis: verum end; then A11: s2 is divergent_to-infty by A5, Th21, Th43; A12: rng s2 c= dom f2 proof let x be real number ; :: according to MEMBERED:def_9 ::_thesis: ( not x in rng s2 or x in dom f2 ) assume x in rng s2 ; ::_thesis: x in dom f2 then ex n being Element of NAT st x = s2 . n by FUNCT_2:113; then x in (dom f1) /\ (left_open_halfline r) by A10; hence x in dom f2 by A4, XBOOLE_0:def_4; ::_thesis: verum end; then A13: lim (f2 /* s2) = lim_in-infty f2 by A2, A11, Def13; A14: rng s2 c= dom f1 proof let x be real number ; :: according to MEMBERED:def_9 ::_thesis: ( not x in rng s2 or x in dom f1 ) assume x in rng s2 ; ::_thesis: x in dom f1 then ex n being Element of NAT st x = s2 . n by FUNCT_2:113; then x in (dom f1) /\ (left_open_halfline r) by A10; hence x in dom f1 by XBOOLE_0:def_4; ::_thesis: verum end; A15: now__::_thesis:_for_n_being_Element_of_NAT_holds_(f1_/*_s2)_._n_<=_(f2_/*_s2)_._n let n be Element of NAT ; ::_thesis: (f1 /* s2) . n <= (f2 /* s2) . n f1 . (s2 . n) <= f2 . (s2 . n) by A4, A10; then (f1 /* s2) . n <= f2 . (s2 . n) by A14, FUNCT_2:108; hence (f1 /* s2) . n <= (f2 /* s2) . n by A12, FUNCT_2:108; ::_thesis: verum end; lim_in-infty f2 = lim_in-infty f2 ; then A16: f2 /* s2 is convergent by A2, A11, A12, Def13; lim_in-infty f1 = lim_in-infty f1 ; then A17: f1 /* s2 is convergent by A1, A11, A14, Def13; lim (f1 /* s2) = lim_in-infty f1 by A1, A11, A14, Def13; hence lim_in-infty f1 <= lim_in-infty f2 by A17, A16, A13, A15, SEQ_2:18; ::_thesis: verum end; supposeA18: ( (dom f2) /\ (left_open_halfline r) c= (dom f1) /\ (left_open_halfline r) & ( for g being Real st g in (dom f2) /\ (left_open_halfline r) holds f1 . g <= f2 . g ) ) ; ::_thesis: lim_in-infty f1 <= lim_in-infty f2 deffunc H1( Element of NAT ) -> Element of REAL = - $1; defpred S1[ Element of NAT , real number ] means ( $2 < - $1 & $2 in (dom f2) /\ (left_open_halfline r) ); consider s1 being Real_Sequence such that A19: for n being Element of NAT holds s1 . n = H1(n) from SEQ_1:sch_1(); A20: now__::_thesis:_for_n_being_Element_of_NAT_ex_g_being_Real_st_S1[n,g] let n be Element of NAT ; ::_thesis: ex g being Real st S1[n,g] 0 <= abs r by COMPLEX1:46; then A21: (- n) - (abs r) <= (- n) - 0 by XREAL_1:13; consider g being Real such that A22: g < (- n) - (abs r) and A23: g in dom f2 by A2, Def9; take g = g; ::_thesis: S1[n,g] ( 0 <= n & - (abs r) <= r ) by ABSVALUE:4, NAT_1:2; then (- (abs r)) - n <= r - 0 by XREAL_1:13; then g < r by A22, XXREAL_0:2; then g in { g2 where g2 is Real : g2 < r } ; then g in left_open_halfline r by XXREAL_1:229; hence S1[n,g] by A22, A23, A21, XBOOLE_0:def_4, XXREAL_0:2; ::_thesis: verum end; consider s2 being Real_Sequence such that A24: for n being Element of NAT holds S1[n,s2 . n] from FUNCT_2:sch_3(A20); now__::_thesis:_for_n_being_Element_of_NAT_holds_s2_._n_<=_s1_._n let n be Element of NAT ; ::_thesis: s2 . n <= s1 . n s2 . n < - n by A24; hence s2 . n <= s1 . n by A19; ::_thesis: verum end; then A25: s2 is divergent_to-infty by A19, Th21, Th43; A26: rng s2 c= dom f1 proof let x be real number ; :: according to MEMBERED:def_9 ::_thesis: ( not x in rng s2 or x in dom f1 ) assume x in rng s2 ; ::_thesis: x in dom f1 then ex n being Element of NAT st x = s2 . n by FUNCT_2:113; then x in (dom f2) /\ (left_open_halfline r) by A24; hence x in dom f1 by A18, XBOOLE_0:def_4; ::_thesis: verum end; then A27: lim (f1 /* s2) = lim_in-infty f1 by A1, A25, Def13; A28: rng s2 c= dom f2 proof let x be real number ; :: according to MEMBERED:def_9 ::_thesis: ( not x in rng s2 or x in dom f2 ) assume x in rng s2 ; ::_thesis: x in dom f2 then ex n being Element of NAT st x = s2 . n by FUNCT_2:113; then x in (dom f2) /\ (left_open_halfline r) by A24; hence x in dom f2 by XBOOLE_0:def_4; ::_thesis: verum end; A29: now__::_thesis:_for_n_being_Element_of_NAT_holds_(f1_/*_s2)_._n_<=_(f2_/*_s2)_._n let n be Element of NAT ; ::_thesis: (f1 /* s2) . n <= (f2 /* s2) . n f1 . (s2 . n) <= f2 . (s2 . n) by A18, A24; then (f1 /* s2) . n <= f2 . (s2 . n) by A26, FUNCT_2:108; hence (f1 /* s2) . n <= (f2 /* s2) . n by A28, FUNCT_2:108; ::_thesis: verum end; lim_in-infty f1 = lim_in-infty f1 ; then A30: f1 /* s2 is convergent by A1, A25, A26, Def13; lim_in-infty f2 = lim_in-infty f2 ; then A31: f2 /* s2 is convergent by A2, A25, A28, Def13; lim (f2 /* s2) = lim_in-infty f2 by A2, A25, A28, Def13; hence lim_in-infty f1 <= lim_in-infty f2 by A31, A30, A27, A29, SEQ_2:18; ::_thesis: verum end; end; end; hence lim_in-infty f1 <= lim_in-infty f2 ; ::_thesis: verum end; theorem :: LIMFUNC1:106 for f being PartFunc of REAL,REAL st ( f is divergent_in+infty_to+infty or f is divergent_in+infty_to-infty ) & ( for r being Real ex g being Real st ( r < g & g in dom f & f . g <> 0 ) ) holds ( f ^ is convergent_in+infty & lim_in+infty (f ^) = 0 ) proof let f be PartFunc of REAL,REAL; ::_thesis: ( ( f is divergent_in+infty_to+infty or f is divergent_in+infty_to-infty ) & ( for r being Real ex g being Real st ( r < g & g in dom f & f . g <> 0 ) ) implies ( f ^ is convergent_in+infty & lim_in+infty (f ^) = 0 ) ) assume that A1: ( f is divergent_in+infty_to+infty or f is divergent_in+infty_to-infty ) and A2: for r being Real ex g being Real st ( r < g & g in dom f & f . g <> 0 ) ; ::_thesis: ( f ^ is convergent_in+infty & lim_in+infty (f ^) = 0 ) A3: dom (f ^) = (dom f) \ (f " {0}) by RFUNCT_1:def_2; A4: now__::_thesis:_for_r_being_Real_ex_g_being_Real_st_ (_r_<_g_&_g_in_dom_(f_^)_) let r be Real; ::_thesis: ex g being Real st ( r < g & g in dom (f ^) ) consider g being Real such that A5: ( r < g & g in dom f ) and A6: f . g <> 0 by A2; take g = g; ::_thesis: ( r < g & g in dom (f ^) ) not f . g in {0} by A6, TARSKI:def_1; then not g in f " {0} by FUNCT_1:def_7; hence ( r < g & g in dom (f ^) ) by A3, A5, XBOOLE_0:def_5; ::_thesis: verum end; now__::_thesis:_(_f_^_is_convergent_in+infty_&_f_^_is_convergent_in+infty_&_lim_in+infty_(f_^)_=_0_) percases ( f is divergent_in+infty_to+infty or f is divergent_in+infty_to-infty ) by A1; supposeA7: f is divergent_in+infty_to+infty ; ::_thesis: ( f ^ is convergent_in+infty & f ^ is convergent_in+infty & lim_in+infty (f ^) = 0 ) A8: now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_divergent_to+infty_&_rng_seq_c=_dom_(f_^)_holds_ (_(f_^)_/*_seq_is_convergent_&_lim_((f_^)_/*_seq)_=_0_) let seq be Real_Sequence; ::_thesis: ( seq is divergent_to+infty & rng seq c= dom (f ^) implies ( (f ^) /* seq is convergent & lim ((f ^) /* seq) = 0 ) ) assume that A9: seq is divergent_to+infty and A10: rng seq c= dom (f ^) ; ::_thesis: ( (f ^) /* seq is convergent & lim ((f ^) /* seq) = 0 ) dom (f ^) c= dom f by A3, XBOOLE_1:36; then rng seq c= dom f by A10, XBOOLE_1:1; then f /* seq is divergent_to+infty by A7, A9, Def7; then ( (f /* seq) " is convergent & lim ((f /* seq) ") = 0 ) by Th34; hence ( (f ^) /* seq is convergent & lim ((f ^) /* seq) = 0 ) by A10, RFUNCT_2:12; ::_thesis: verum end; hence f ^ is convergent_in+infty by A4, Def6; ::_thesis: ( f ^ is convergent_in+infty & lim_in+infty (f ^) = 0 ) hence ( f ^ is convergent_in+infty & lim_in+infty (f ^) = 0 ) by A8, Def12; ::_thesis: verum end; supposeA11: f is divergent_in+infty_to-infty ; ::_thesis: ( f ^ is convergent_in+infty & f ^ is convergent_in+infty & lim_in+infty (f ^) = 0 ) A12: now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_divergent_to+infty_&_rng_seq_c=_dom_(f_^)_holds_ (_(f_^)_/*_seq_is_convergent_&_lim_((f_^)_/*_seq)_=_0_) let seq be Real_Sequence; ::_thesis: ( seq is divergent_to+infty & rng seq c= dom (f ^) implies ( (f ^) /* seq is convergent & lim ((f ^) /* seq) = 0 ) ) assume that A13: seq is divergent_to+infty and A14: rng seq c= dom (f ^) ; ::_thesis: ( (f ^) /* seq is convergent & lim ((f ^) /* seq) = 0 ) dom (f ^) c= dom f by A3, XBOOLE_1:36; then rng seq c= dom f by A14, XBOOLE_1:1; then f /* seq is divergent_to-infty by A11, A13, Def8; then ( (f /* seq) " is convergent & lim ((f /* seq) ") = 0 ) by Th34; hence ( (f ^) /* seq is convergent & lim ((f ^) /* seq) = 0 ) by A14, RFUNCT_2:12; ::_thesis: verum end; hence f ^ is convergent_in+infty by A4, Def6; ::_thesis: ( f ^ is convergent_in+infty & lim_in+infty (f ^) = 0 ) hence ( f ^ is convergent_in+infty & lim_in+infty (f ^) = 0 ) by A12, Def12; ::_thesis: verum end; end; end; hence ( f ^ is convergent_in+infty & lim_in+infty (f ^) = 0 ) ; ::_thesis: verum end; theorem :: LIMFUNC1:107 for f being PartFunc of REAL,REAL st ( f is divergent_in-infty_to+infty or f is divergent_in-infty_to-infty ) & ( for r being Real ex g being Real st ( g < r & g in dom f & f . g <> 0 ) ) holds ( f ^ is convergent_in-infty & lim_in-infty (f ^) = 0 ) proof let f be PartFunc of REAL,REAL; ::_thesis: ( ( f is divergent_in-infty_to+infty or f is divergent_in-infty_to-infty ) & ( for r being Real ex g being Real st ( g < r & g in dom f & f . g <> 0 ) ) implies ( f ^ is convergent_in-infty & lim_in-infty (f ^) = 0 ) ) assume that A1: ( f is divergent_in-infty_to+infty or f is divergent_in-infty_to-infty ) and A2: for r being Real ex g being Real st ( g < r & g in dom f & f . g <> 0 ) ; ::_thesis: ( f ^ is convergent_in-infty & lim_in-infty (f ^) = 0 ) A3: dom (f ^) = (dom f) \ (f " {0}) by RFUNCT_1:def_2; A4: now__::_thesis:_for_r_being_Real_ex_g_being_Real_st_ (_g_<_r_&_g_in_dom_(f_^)_) let r be Real; ::_thesis: ex g being Real st ( g < r & g in dom (f ^) ) consider g being Real such that A5: ( g < r & g in dom f ) and A6: f . g <> 0 by A2; take g = g; ::_thesis: ( g < r & g in dom (f ^) ) not f . g in {0} by A6, TARSKI:def_1; then not g in f " {0} by FUNCT_1:def_7; hence ( g < r & g in dom (f ^) ) by A3, A5, XBOOLE_0:def_5; ::_thesis: verum end; now__::_thesis:_(_f_^_is_convergent_in-infty_&_f_^_is_convergent_in-infty_&_lim_in-infty_(f_^)_=_0_) percases ( f is divergent_in-infty_to+infty or f is divergent_in-infty_to-infty ) by A1; supposeA7: f is divergent_in-infty_to+infty ; ::_thesis: ( f ^ is convergent_in-infty & f ^ is convergent_in-infty & lim_in-infty (f ^) = 0 ) A8: now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_divergent_to-infty_&_rng_seq_c=_dom_(f_^)_holds_ (_(f_^)_/*_seq_is_convergent_&_lim_((f_^)_/*_seq)_=_0_) let seq be Real_Sequence; ::_thesis: ( seq is divergent_to-infty & rng seq c= dom (f ^) implies ( (f ^) /* seq is convergent & lim ((f ^) /* seq) = 0 ) ) assume that A9: seq is divergent_to-infty and A10: rng seq c= dom (f ^) ; ::_thesis: ( (f ^) /* seq is convergent & lim ((f ^) /* seq) = 0 ) dom (f ^) c= dom f by A3, XBOOLE_1:36; then rng seq c= dom f by A10, XBOOLE_1:1; then f /* seq is divergent_to+infty by A7, A9, Def10; then ( (f /* seq) " is convergent & lim ((f /* seq) ") = 0 ) by Th34; hence ( (f ^) /* seq is convergent & lim ((f ^) /* seq) = 0 ) by A10, RFUNCT_2:12; ::_thesis: verum end; hence f ^ is convergent_in-infty by A4, Def9; ::_thesis: ( f ^ is convergent_in-infty & lim_in-infty (f ^) = 0 ) hence ( f ^ is convergent_in-infty & lim_in-infty (f ^) = 0 ) by A8, Def13; ::_thesis: verum end; supposeA11: f is divergent_in-infty_to-infty ; ::_thesis: ( f ^ is convergent_in-infty & f ^ is convergent_in-infty & lim_in-infty (f ^) = 0 ) A12: now__::_thesis:_for_seq_being_Real_Sequence_st_seq_is_divergent_to-infty_&_rng_seq_c=_dom_(f_^)_holds_ (_(f_^)_/*_seq_is_convergent_&_lim_((f_^)_/*_seq)_=_0_) let seq be Real_Sequence; ::_thesis: ( seq is divergent_to-infty & rng seq c= dom (f ^) implies ( (f ^) /* seq is convergent & lim ((f ^) /* seq) = 0 ) ) assume that A13: seq is divergent_to-infty and A14: rng seq c= dom (f ^) ; ::_thesis: ( (f ^) /* seq is convergent & lim ((f ^) /* seq) = 0 ) dom (f ^) c= dom f by A3, XBOOLE_1:36; then rng seq c= dom f by A14, XBOOLE_1:1; then f /* seq is divergent_to-infty by A11, A13, Def11; then ( (f /* seq) " is convergent & lim ((f /* seq) ") = 0 ) by Th34; hence ( (f ^) /* seq is convergent & lim ((f ^) /* seq) = 0 ) by A14, RFUNCT_2:12; ::_thesis: verum end; hence f ^ is convergent_in-infty by A4, Def9; ::_thesis: ( f ^ is convergent_in-infty & lim_in-infty (f ^) = 0 ) hence ( f ^ is convergent_in-infty & lim_in-infty (f ^) = 0 ) by A12, Def13; ::_thesis: verum end; end; end; hence ( f ^ is convergent_in-infty & lim_in-infty (f ^) = 0 ) ; ::_thesis: verum end; theorem :: LIMFUNC1:108 for f being PartFunc of REAL,REAL st f is convergent_in+infty & lim_in+infty f = 0 & ( for r being Real ex g being Real st ( r < g & g in dom f & f . g <> 0 ) ) & ex r being Real st for g being Real st g in (dom f) /\ (right_open_halfline r) holds 0 <= f . g holds f ^ is divergent_in+infty_to+infty proof let f be PartFunc of REAL,REAL; ::_thesis: ( f is convergent_in+infty & lim_in+infty f = 0 & ( for r being Real ex g being Real st ( r < g & g in dom f & f . g <> 0 ) ) & ex r being Real st for g being Real st g in (dom f) /\ (right_open_halfline r) holds 0 <= f . g implies f ^ is divergent_in+infty_to+infty ) assume that A1: ( f is convergent_in+infty & lim_in+infty f = 0 ) and A2: for r being Real ex g being Real st ( r < g & g in dom f & f . g <> 0 ) ; ::_thesis: ( for r being Real ex g being Real st ( g in (dom f) /\ (right_open_halfline r) & not 0 <= f . g ) or f ^ is divergent_in+infty_to+infty ) given r being Real such that A3: for g being Real st g in (dom f) /\ (right_open_halfline r) holds 0 <= f . g ; ::_thesis: f ^ is divergent_in+infty_to+infty thus for r1 being Real ex g1 being Real st ( r1 < g1 & g1 in dom (f ^) ) :: according to LIMFUNC1:def_7 ::_thesis: for seq being Real_Sequence st seq is divergent_to+infty & rng seq c= dom (f ^) holds (f ^) /* seq is divergent_to+infty proof let r1 be Real; ::_thesis: ex g1 being Real st ( r1 < g1 & g1 in dom (f ^) ) consider g1 being Real such that A4: r1 < g1 and A5: g1 in dom f and A6: f . g1 <> 0 by A2; take g1 ; ::_thesis: ( r1 < g1 & g1 in dom (f ^) ) thus r1 < g1 by A4; ::_thesis: g1 in dom (f ^) not f . g1 in {0} by A6, TARSKI:def_1; then not g1 in f " {0} by FUNCT_1:def_7; then g1 in (dom f) \ (f " {0}) by A5, XBOOLE_0:def_5; hence g1 in dom (f ^) by RFUNCT_1:def_2; ::_thesis: verum end; let s be Real_Sequence; ::_thesis: ( s is divergent_to+infty & rng s c= dom (f ^) implies (f ^) /* s is divergent_to+infty ) assume that A7: s is divergent_to+infty and A8: rng s c= dom (f ^) ; ::_thesis: (f ^) /* s is divergent_to+infty consider k being Element of NAT such that A9: for n being Element of NAT st k <= n holds r < s . n by A7, Def4; A10: rng (s ^\ k) c= rng s by VALUED_0:21; dom (f ^) = (dom f) \ (f " {0}) by RFUNCT_1:def_2; then A11: dom (f ^) c= dom f by XBOOLE_1:36; then A12: rng s c= dom f by A8, XBOOLE_1:1; then A13: rng (s ^\ k) c= dom f by A10, XBOOLE_1:1; A14: f /* (s ^\ k) is non-zero by A8, A10, RFUNCT_2:11, XBOOLE_1:1; now__::_thesis:_for_n_being_Element_of_NAT_holds_0_<_(f_/*_(s_^\_k))_._n let n be Element of NAT ; ::_thesis: 0 < (f /* (s ^\ k)) . n r < s . (n + k) by A9, NAT_1:12; then r < (s ^\ k) . n by NAT_1:def_3; then (s ^\ k) . n in { g2 where g2 is Real : r < g2 } ; then ( (s ^\ k) . n in rng (s ^\ k) & (s ^\ k) . n in right_open_halfline r ) by VALUED_0:28, XXREAL_1:230; then (s ^\ k) . n in (dom f) /\ (right_open_halfline r) by A13, XBOOLE_0:def_4; then A15: 0 <= f . ((s ^\ k) . n) by A3; (f /* (s ^\ k)) . n <> 0 by A14, SEQ_1:5; hence 0 < (f /* (s ^\ k)) . n by A12, A10, A15, FUNCT_2:108, XBOOLE_1:1; ::_thesis: verum end; then A16: for n being Element of NAT st 0 <= n holds 0 < (f /* (s ^\ k)) . n ; s ^\ k is divergent_to+infty by A7, Th26; then ( f /* (s ^\ k) is convergent & lim (f /* (s ^\ k)) = 0 ) by A1, A13, Def12; then A17: (f /* (s ^\ k)) " is divergent_to+infty by A16, Th35; (f /* (s ^\ k)) " = ((f /* s) ^\ k) " by A8, A11, VALUED_0:27, XBOOLE_1:1 .= ((f /* s) ") ^\ k by SEQM_3:18 .= ((f ^) /* s) ^\ k by A8, RFUNCT_2:12 ; hence (f ^) /* s is divergent_to+infty by A17, Th7; ::_thesis: verum end; theorem :: LIMFUNC1:109 for f being PartFunc of REAL,REAL st f is convergent_in+infty & lim_in+infty f = 0 & ( for r being Real ex g being Real st ( r < g & g in dom f & f . g <> 0 ) ) & ex r being Real st for g being Real st g in (dom f) /\ (right_open_halfline r) holds f . g <= 0 holds f ^ is divergent_in+infty_to-infty proof let f be PartFunc of REAL,REAL; ::_thesis: ( f is convergent_in+infty & lim_in+infty f = 0 & ( for r being Real ex g being Real st ( r < g & g in dom f & f . g <> 0 ) ) & ex r being Real st for g being Real st g in (dom f) /\ (right_open_halfline r) holds f . g <= 0 implies f ^ is divergent_in+infty_to-infty ) assume that A1: ( f is convergent_in+infty & lim_in+infty f = 0 ) and A2: for r being Real ex g being Real st ( r < g & g in dom f & f . g <> 0 ) ; ::_thesis: ( for r being Real ex g being Real st ( g in (dom f) /\ (right_open_halfline r) & not f . g <= 0 ) or f ^ is divergent_in+infty_to-infty ) given r being Real such that A3: for g being Real st g in (dom f) /\ (right_open_halfline r) holds f . g <= 0 ; ::_thesis: f ^ is divergent_in+infty_to-infty thus for r1 being Real ex g1 being Real st ( r1 < g1 & g1 in dom (f ^) ) :: according to LIMFUNC1:def_8 ::_thesis: for seq being Real_Sequence st seq is divergent_to+infty & rng seq c= dom (f ^) holds (f ^) /* seq is divergent_to-infty proof let r1 be Real; ::_thesis: ex g1 being Real st ( r1 < g1 & g1 in dom (f ^) ) consider g1 being Real such that A4: r1 < g1 and A5: g1 in dom f and A6: f . g1 <> 0 by A2; take g1 ; ::_thesis: ( r1 < g1 & g1 in dom (f ^) ) thus r1 < g1 by A4; ::_thesis: g1 in dom (f ^) not f . g1 in {0} by A6, TARSKI:def_1; then not g1 in f " {0} by FUNCT_1:def_7; then g1 in (dom f) \ (f " {0}) by A5, XBOOLE_0:def_5; hence g1 in dom (f ^) by RFUNCT_1:def_2; ::_thesis: verum end; let s be Real_Sequence; ::_thesis: ( s is divergent_to+infty & rng s c= dom (f ^) implies (f ^) /* s is divergent_to-infty ) assume that A7: s is divergent_to+infty and A8: rng s c= dom (f ^) ; ::_thesis: (f ^) /* s is divergent_to-infty consider k being Element of NAT such that A9: for n being Element of NAT st k <= n holds r < s . n by A7, Def4; A10: rng (s ^\ k) c= rng s by VALUED_0:21; dom (f ^) = (dom f) \ (f " {0}) by RFUNCT_1:def_2; then A11: dom (f ^) c= dom f by XBOOLE_1:36; then A12: rng s c= dom f by A8, XBOOLE_1:1; then A13: rng (s ^\ k) c= dom f by A10, XBOOLE_1:1; A14: f /* (s ^\ k) is non-zero by A8, A10, RFUNCT_2:11, XBOOLE_1:1; now__::_thesis:_for_n_being_Element_of_NAT_holds_(f_/*_(s_^\_k))_._n_<_0 let n be Element of NAT ; ::_thesis: (f /* (s ^\ k)) . n < 0 r < s . (n + k) by A9, NAT_1:12; then r < (s ^\ k) . n by NAT_1:def_3; then (s ^\ k) . n in { g2 where g2 is Real : r < g2 } ; then ( (s ^\ k) . n in rng (s ^\ k) & (s ^\ k) . n in right_open_halfline r ) by VALUED_0:28, XXREAL_1:230; then (s ^\ k) . n in (dom f) /\ (right_open_halfline r) by A13, XBOOLE_0:def_4; then A15: f . ((s ^\ k) . n) <= 0 by A3; (f /* (s ^\ k)) . n <> 0 by A14, SEQ_1:5; hence (f /* (s ^\ k)) . n < 0 by A12, A10, A15, FUNCT_2:108, XBOOLE_1:1; ::_thesis: verum end; then A16: for n being Element of NAT st 0 <= n holds (f /* (s ^\ k)) . n < 0 ; s ^\ k is divergent_to+infty by A7, Th26; then ( f /* (s ^\ k) is convergent & lim (f /* (s ^\ k)) = 0 ) by A1, A13, Def12; then A17: (f /* (s ^\ k)) " is divergent_to-infty by A16, Th36; (f /* (s ^\ k)) " = ((f /* s) ^\ k) " by A8, A11, VALUED_0:27, XBOOLE_1:1 .= ((f /* s) ") ^\ k by SEQM_3:18 .= ((f ^) /* s) ^\ k by A8, RFUNCT_2:12 ; hence (f ^) /* s is divergent_to-infty by A17, Th7; ::_thesis: verum end; theorem :: LIMFUNC1:110 for f being PartFunc of REAL,REAL st f is convergent_in-infty & lim_in-infty f = 0 & ( for r being Real ex g being Real st ( g < r & g in dom f & f . g <> 0 ) ) & ex r being Real st for g being Real st g in (dom f) /\ (left_open_halfline r) holds 0 <= f . g holds f ^ is divergent_in-infty_to+infty proof let f be PartFunc of REAL,REAL; ::_thesis: ( f is convergent_in-infty & lim_in-infty f = 0 & ( for r being Real ex g being Real st ( g < r & g in dom f & f . g <> 0 ) ) & ex r being Real st for g being Real st g in (dom f) /\ (left_open_halfline r) holds 0 <= f . g implies f ^ is divergent_in-infty_to+infty ) assume that A1: ( f is convergent_in-infty & lim_in-infty f = 0 ) and A2: for r being Real ex g being Real st ( g < r & g in dom f & f . g <> 0 ) ; ::_thesis: ( for r being Real ex g being Real st ( g in (dom f) /\ (left_open_halfline r) & not 0 <= f . g ) or f ^ is divergent_in-infty_to+infty ) given r being Real such that A3: for g being Real st g in (dom f) /\ (left_open_halfline r) holds 0 <= f . g ; ::_thesis: f ^ is divergent_in-infty_to+infty thus for r1 being Real ex g1 being Real st ( g1 < r1 & g1 in dom (f ^) ) :: according to LIMFUNC1:def_10 ::_thesis: for seq being Real_Sequence st seq is divergent_to-infty & rng seq c= dom (f ^) holds (f ^) /* seq is divergent_to+infty proof let r1 be Real; ::_thesis: ex g1 being Real st ( g1 < r1 & g1 in dom (f ^) ) consider g1 being Real such that A4: g1 < r1 and A5: g1 in dom f and A6: f . g1 <> 0 by A2; take g1 ; ::_thesis: ( g1 < r1 & g1 in dom (f ^) ) thus g1 < r1 by A4; ::_thesis: g1 in dom (f ^) not f . g1 in {0} by A6, TARSKI:def_1; then not g1 in f " {0} by FUNCT_1:def_7; then g1 in (dom f) \ (f " {0}) by A5, XBOOLE_0:def_5; hence g1 in dom (f ^) by RFUNCT_1:def_2; ::_thesis: verum end; let s be Real_Sequence; ::_thesis: ( s is divergent_to-infty & rng s c= dom (f ^) implies (f ^) /* s is divergent_to+infty ) assume that A7: s is divergent_to-infty and A8: rng s c= dom (f ^) ; ::_thesis: (f ^) /* s is divergent_to+infty consider k being Element of NAT such that A9: for n being Element of NAT st k <= n holds s . n < r by A7, Def5; A10: rng (s ^\ k) c= rng s by VALUED_0:21; dom (f ^) = (dom f) \ (f " {0}) by RFUNCT_1:def_2; then A11: dom (f ^) c= dom f by XBOOLE_1:36; then A12: rng s c= dom f by A8, XBOOLE_1:1; then A13: rng (s ^\ k) c= dom f by A10, XBOOLE_1:1; A14: f /* (s ^\ k) is non-zero by A8, A10, RFUNCT_2:11, XBOOLE_1:1; now__::_thesis:_for_n_being_Element_of_NAT_holds_0_<_(f_/*_(s_^\_k))_._n let n be Element of NAT ; ::_thesis: 0 < (f /* (s ^\ k)) . n s . (n + k) < r by A9, NAT_1:12; then (s ^\ k) . n < r by NAT_1:def_3; then (s ^\ k) . n in { g2 where g2 is Real : g2 < r } ; then ( (s ^\ k) . n in rng (s ^\ k) & (s ^\ k) . n in left_open_halfline r ) by VALUED_0:28, XXREAL_1:229; then (s ^\ k) . n in (dom f) /\ (left_open_halfline r) by A13, XBOOLE_0:def_4; then A15: 0 <= f . ((s ^\ k) . n) by A3; 0 <> (f /* (s ^\ k)) . n by A14, SEQ_1:5; hence 0 < (f /* (s ^\ k)) . n by A12, A10, A15, FUNCT_2:108, XBOOLE_1:1; ::_thesis: verum end; then A16: for n being Element of NAT st 0 <= n holds 0 < (f /* (s ^\ k)) . n ; s ^\ k is divergent_to-infty by A7, Th27; then ( f /* (s ^\ k) is convergent & lim (f /* (s ^\ k)) = 0 ) by A1, A13, Def13; then A17: (f /* (s ^\ k)) " is divergent_to+infty by A16, Th35; (f /* (s ^\ k)) " = ((f /* s) ^\ k) " by A8, A11, VALUED_0:27, XBOOLE_1:1 .= ((f /* s) ") ^\ k by SEQM_3:18 .= ((f ^) /* s) ^\ k by A8, RFUNCT_2:12 ; hence (f ^) /* s is divergent_to+infty by A17, Th7; ::_thesis: verum end; theorem :: LIMFUNC1:111 for f being PartFunc of REAL,REAL st f is convergent_in-infty & lim_in-infty f = 0 & ( for r being Real ex g being Real st ( g < r & g in dom f & f . g <> 0 ) ) & ex r being Real st for g being Real st g in (dom f) /\ (left_open_halfline r) holds f . g <= 0 holds f ^ is divergent_in-infty_to-infty proof let f be PartFunc of REAL,REAL; ::_thesis: ( f is convergent_in-infty & lim_in-infty f = 0 & ( for r being Real ex g being Real st ( g < r & g in dom f & f . g <> 0 ) ) & ex r being Real st for g being Real st g in (dom f) /\ (left_open_halfline r) holds f . g <= 0 implies f ^ is divergent_in-infty_to-infty ) assume that A1: ( f is convergent_in-infty & lim_in-infty f = 0 ) and A2: for r being Real ex g being Real st ( g < r & g in dom f & f . g <> 0 ) ; ::_thesis: ( for r being Real ex g being Real st ( g in (dom f) /\ (left_open_halfline r) & not f . g <= 0 ) or f ^ is divergent_in-infty_to-infty ) given r being Real such that A3: for g being Real st g in (dom f) /\ (left_open_halfline r) holds f . g <= 0 ; ::_thesis: f ^ is divergent_in-infty_to-infty thus for r1 being Real ex g1 being Real st ( g1 < r1 & g1 in dom (f ^) ) :: according to LIMFUNC1:def_11 ::_thesis: for seq being Real_Sequence st seq is divergent_to-infty & rng seq c= dom (f ^) holds (f ^) /* seq is divergent_to-infty proof let r1 be Real; ::_thesis: ex g1 being Real st ( g1 < r1 & g1 in dom (f ^) ) consider g1 being Real such that A4: g1 < r1 and A5: g1 in dom f and A6: f . g1 <> 0 by A2; take g1 ; ::_thesis: ( g1 < r1 & g1 in dom (f ^) ) thus g1 < r1 by A4; ::_thesis: g1 in dom (f ^) not f . g1 in {0} by A6, TARSKI:def_1; then not g1 in f " {0} by FUNCT_1:def_7; then g1 in (dom f) \ (f " {0}) by A5, XBOOLE_0:def_5; hence g1 in dom (f ^) by RFUNCT_1:def_2; ::_thesis: verum end; let s be Real_Sequence; ::_thesis: ( s is divergent_to-infty & rng s c= dom (f ^) implies (f ^) /* s is divergent_to-infty ) assume that A7: s is divergent_to-infty and A8: rng s c= dom (f ^) ; ::_thesis: (f ^) /* s is divergent_to-infty consider k being Element of NAT such that A9: for n being Element of NAT st k <= n holds s . n < r by A7, Def5; A10: rng (s ^\ k) c= rng s by VALUED_0:21; dom (f ^) = (dom f) \ (f " {0}) by RFUNCT_1:def_2; then A11: dom (f ^) c= dom f by XBOOLE_1:36; then A12: rng s c= dom f by A8, XBOOLE_1:1; then A13: rng (s ^\ k) c= dom f by A10, XBOOLE_1:1; A14: f /* (s ^\ k) is non-zero by A8, A10, RFUNCT_2:11, XBOOLE_1:1; now__::_thesis:_for_n_being_Element_of_NAT_holds_(f_/*_(s_^\_k))_._n_<_0 let n be Element of NAT ; ::_thesis: (f /* (s ^\ k)) . n < 0 s . (n + k) < r by A9, NAT_1:12; then (s ^\ k) . n < r by NAT_1:def_3; then (s ^\ k) . n in { g2 where g2 is Real : g2 < r } ; then ( (s ^\ k) . n in rng (s ^\ k) & (s ^\ k) . n in left_open_halfline r ) by VALUED_0:28, XXREAL_1:229; then (s ^\ k) . n in (dom f) /\ (left_open_halfline r) by A13, XBOOLE_0:def_4; then A15: f . ((s ^\ k) . n) <= 0 by A3; (f /* (s ^\ k)) . n <> 0 by A14, SEQ_1:5; hence (f /* (s ^\ k)) . n < 0 by A12, A10, A15, FUNCT_2:108, XBOOLE_1:1; ::_thesis: verum end; then A16: for n being Element of NAT st 0 <= n holds (f /* (s ^\ k)) . n < 0 ; s ^\ k is divergent_to-infty by A7, Th27; then ( f /* (s ^\ k) is convergent & lim (f /* (s ^\ k)) = 0 ) by A1, A13, Def13; then A17: (f /* (s ^\ k)) " is divergent_to-infty by A16, Th36; (f /* (s ^\ k)) " = ((f /* s) ^\ k) " by A8, A11, VALUED_0:27, XBOOLE_1:1 .= ((f /* s) ") ^\ k by SEQM_3:18 .= ((f ^) /* s) ^\ k by A8, RFUNCT_2:12 ; hence (f ^) /* s is divergent_to-infty by A17, Th7; ::_thesis: verum end; theorem :: LIMFUNC1:112 for f being PartFunc of REAL,REAL st f is convergent_in+infty & lim_in+infty f = 0 & ex r being Real st for g being Real st g in (dom f) /\ (right_open_halfline r) holds 0 < f . g holds f ^ is divergent_in+infty_to+infty proof let f be PartFunc of REAL,REAL; ::_thesis: ( f is convergent_in+infty & lim_in+infty f = 0 & ex r being Real st for g being Real st g in (dom f) /\ (right_open_halfline r) holds 0 < f . g implies f ^ is divergent_in+infty_to+infty ) assume that A1: f is convergent_in+infty and A2: lim_in+infty f = 0 ; ::_thesis: ( for r being Real ex g being Real st ( g in (dom f) /\ (right_open_halfline r) & not 0 < f . g ) or f ^ is divergent_in+infty_to+infty ) given r being Real such that A3: for g being Real st g in (dom f) /\ (right_open_halfline r) holds 0 < f . g ; ::_thesis: f ^ is divergent_in+infty_to+infty thus for r1 being Real ex g1 being Real st ( r1 < g1 & g1 in dom (f ^) ) :: according to LIMFUNC1:def_7 ::_thesis: for seq being Real_Sequence st seq is divergent_to+infty & rng seq c= dom (f ^) holds (f ^) /* seq is divergent_to+infty proof let r1 be Real; ::_thesis: ex g1 being Real st ( r1 < g1 & g1 in dom (f ^) ) consider g1 being Real such that A4: r1 < g1 and g1 in dom f by A1, Def6; now__::_thesis:_ex_g2_being_Real_st_ (_r1_<_g2_&_g2_in_dom_(f_^)_) percases ( g1 <= r or r <= g1 ) ; supposeA5: g1 <= r ; ::_thesis: ex g2 being Real st ( r1 < g2 & g2 in dom (f ^) ) consider g2 being Real such that A6: r < g2 and A7: g2 in dom f by A1, Def6; take g2 = g2; ::_thesis: ( r1 < g2 & g2 in dom (f ^) ) g1 < g2 by A5, A6, XXREAL_0:2; hence r1 < g2 by A4, XXREAL_0:2; ::_thesis: g2 in dom (f ^) g2 in { r2 where r2 is Real : r < r2 } by A6; then g2 in right_open_halfline r by XXREAL_1:230; then g2 in (dom f) /\ (right_open_halfline r) by A7, XBOOLE_0:def_4; then 0 <> f . g2 by A3; then not f . g2 in {0} by TARSKI:def_1; then not g2 in f " {0} by FUNCT_1:def_7; then g2 in (dom f) \ (f " {0}) by A7, XBOOLE_0:def_5; hence g2 in dom (f ^) by RFUNCT_1:def_2; ::_thesis: verum end; supposeA8: r <= g1 ; ::_thesis: ex g2 being Real st ( r1 < g2 & g2 in dom (f ^) ) consider g2 being Real such that A9: g1 < g2 and A10: g2 in dom f by A1, Def6; take g2 = g2; ::_thesis: ( r1 < g2 & g2 in dom (f ^) ) thus r1 < g2 by A4, A9, XXREAL_0:2; ::_thesis: g2 in dom (f ^) r < g2 by A8, A9, XXREAL_0:2; then g2 in { r2 where r2 is Real : r < r2 } ; then g2 in right_open_halfline r by XXREAL_1:230; then g2 in (dom f) /\ (right_open_halfline r) by A10, XBOOLE_0:def_4; then 0 <> f . g2 by A3; then not f . g2 in {0} by TARSKI:def_1; then not g2 in f " {0} by FUNCT_1:def_7; then g2 in (dom f) \ (f " {0}) by A10, XBOOLE_0:def_5; hence g2 in dom (f ^) by RFUNCT_1:def_2; ::_thesis: verum end; end; end; hence ex g1 being Real st ( r1 < g1 & g1 in dom (f ^) ) ; ::_thesis: verum end; let s be Real_Sequence; ::_thesis: ( s is divergent_to+infty & rng s c= dom (f ^) implies (f ^) /* s is divergent_to+infty ) assume that A11: s is divergent_to+infty and A12: rng s c= dom (f ^) ; ::_thesis: (f ^) /* s is divergent_to+infty consider k being Element of NAT such that A13: for n being Element of NAT st k <= n holds r < s . n by A11, Def4; A14: rng (s ^\ k) c= rng s by VALUED_0:21; dom (f ^) = (dom f) \ (f " {0}) by RFUNCT_1:def_2; then A15: dom (f ^) c= dom f by XBOOLE_1:36; then A16: rng s c= dom f by A12, XBOOLE_1:1; then A17: rng (s ^\ k) c= dom f by A14, XBOOLE_1:1; now__::_thesis:_for_n_being_Element_of_NAT_holds_0_<_(f_/*_(s_^\_k))_._n let n be Element of NAT ; ::_thesis: 0 < (f /* (s ^\ k)) . n r < s . (n + k) by A13, NAT_1:12; then r < (s ^\ k) . n by NAT_1:def_3; then (s ^\ k) . n in { g2 where g2 is Real : r < g2 } ; then ( (s ^\ k) . n in rng (s ^\ k) & (s ^\ k) . n in right_open_halfline r ) by VALUED_0:28, XXREAL_1:230; then (s ^\ k) . n in (dom f) /\ (right_open_halfline r) by A17, XBOOLE_0:def_4; then 0 < f . ((s ^\ k) . n) by A3; hence 0 < (f /* (s ^\ k)) . n by A16, A14, FUNCT_2:108, XBOOLE_1:1; ::_thesis: verum end; then A18: for n being Element of NAT st 0 <= n holds 0 < (f /* (s ^\ k)) . n ; s ^\ k is divergent_to+infty by A11, Th26; then ( f /* (s ^\ k) is convergent & lim (f /* (s ^\ k)) = 0 ) by A1, A2, A17, Def12; then A19: (f /* (s ^\ k)) " is divergent_to+infty by A18, Th35; (f /* (s ^\ k)) " = ((f /* s) ^\ k) " by A12, A15, VALUED_0:27, XBOOLE_1:1 .= ((f /* s) ") ^\ k by SEQM_3:18 .= ((f ^) /* s) ^\ k by A12, RFUNCT_2:12 ; hence (f ^) /* s is divergent_to+infty by A19, Th7; ::_thesis: verum end; theorem :: LIMFUNC1:113 for f being PartFunc of REAL,REAL st f is convergent_in+infty & lim_in+infty f = 0 & ex r being Real st for g being Real st g in (dom f) /\ (right_open_halfline r) holds f . g < 0 holds f ^ is divergent_in+infty_to-infty proof let f be PartFunc of REAL,REAL; ::_thesis: ( f is convergent_in+infty & lim_in+infty f = 0 & ex r being Real st for g being Real st g in (dom f) /\ (right_open_halfline r) holds f . g < 0 implies f ^ is divergent_in+infty_to-infty ) assume that A1: f is convergent_in+infty and A2: lim_in+infty f = 0 ; ::_thesis: ( for r being Real ex g being Real st ( g in (dom f) /\ (right_open_halfline r) & not f . g < 0 ) or f ^ is divergent_in+infty_to-infty ) given r being Real such that A3: for g being Real st g in (dom f) /\ (right_open_halfline r) holds f . g < 0 ; ::_thesis: f ^ is divergent_in+infty_to-infty thus for r1 being Real ex g1 being Real st ( r1 < g1 & g1 in dom (f ^) ) :: according to LIMFUNC1:def_8 ::_thesis: for seq being Real_Sequence st seq is divergent_to+infty & rng seq c= dom (f ^) holds (f ^) /* seq is divergent_to-infty proof let r1 be Real; ::_thesis: ex g1 being Real st ( r1 < g1 & g1 in dom (f ^) ) consider g1 being Real such that A4: r1 < g1 and g1 in dom f by A1, Def6; now__::_thesis:_ex_g2_being_Real_st_ (_r1_<_g2_&_g2_in_dom_(f_^)_) percases ( g1 <= r or r <= g1 ) ; supposeA5: g1 <= r ; ::_thesis: ex g2 being Real st ( r1 < g2 & g2 in dom (f ^) ) consider g2 being Real such that A6: r < g2 and A7: g2 in dom f by A1, Def6; take g2 = g2; ::_thesis: ( r1 < g2 & g2 in dom (f ^) ) g1 < g2 by A5, A6, XXREAL_0:2; hence r1 < g2 by A4, XXREAL_0:2; ::_thesis: g2 in dom (f ^) g2 in { r2 where r2 is Real : r < r2 } by A6; then g2 in right_open_halfline r by XXREAL_1:230; then g2 in (dom f) /\ (right_open_halfline r) by A7, XBOOLE_0:def_4; then 0 <> f . g2 by A3; then not f . g2 in {0} by TARSKI:def_1; then not g2 in f " {0} by FUNCT_1:def_7; then g2 in (dom f) \ (f " {0}) by A7, XBOOLE_0:def_5; hence g2 in dom (f ^) by RFUNCT_1:def_2; ::_thesis: verum end; supposeA8: r <= g1 ; ::_thesis: ex g2 being Real st ( r1 < g2 & g2 in dom (f ^) ) consider g2 being Real such that A9: g1 < g2 and A10: g2 in dom f by A1, Def6; take g2 = g2; ::_thesis: ( r1 < g2 & g2 in dom (f ^) ) thus r1 < g2 by A4, A9, XXREAL_0:2; ::_thesis: g2 in dom (f ^) r < g2 by A8, A9, XXREAL_0:2; then g2 in { r2 where r2 is Real : r < r2 } ; then g2 in right_open_halfline r by XXREAL_1:230; then g2 in (dom f) /\ (right_open_halfline r) by A10, XBOOLE_0:def_4; then 0 <> f . g2 by A3; then not f . g2 in {0} by TARSKI:def_1; then not g2 in f " {0} by FUNCT_1:def_7; then g2 in (dom f) \ (f " {0}) by A10, XBOOLE_0:def_5; hence g2 in dom (f ^) by RFUNCT_1:def_2; ::_thesis: verum end; end; end; hence ex g1 being Real st ( r1 < g1 & g1 in dom (f ^) ) ; ::_thesis: verum end; let s be Real_Sequence; ::_thesis: ( s is divergent_to+infty & rng s c= dom (f ^) implies (f ^) /* s is divergent_to-infty ) assume that A11: s is divergent_to+infty and A12: rng s c= dom (f ^) ; ::_thesis: (f ^) /* s is divergent_to-infty consider k being Element of NAT such that A13: for n being Element of NAT st k <= n holds r < s . n by A11, Def4; A14: rng (s ^\ k) c= rng s by VALUED_0:21; dom (f ^) = (dom f) \ (f " {0}) by RFUNCT_1:def_2; then A15: dom (f ^) c= dom f by XBOOLE_1:36; then A16: rng s c= dom f by A12, XBOOLE_1:1; then A17: rng (s ^\ k) c= dom f by A14, XBOOLE_1:1; now__::_thesis:_for_n_being_Element_of_NAT_holds_(f_/*_(s_^\_k))_._n_<_0 let n be Element of NAT ; ::_thesis: (f /* (s ^\ k)) . n < 0 r < s . (n + k) by A13, NAT_1:12; then r < (s ^\ k) . n by NAT_1:def_3; then (s ^\ k) . n in { g2 where g2 is Real : r < g2 } ; then ( (s ^\ k) . n in rng (s ^\ k) & (s ^\ k) . n in right_open_halfline r ) by VALUED_0:28, XXREAL_1:230; then (s ^\ k) . n in (dom f) /\ (right_open_halfline r) by A17, XBOOLE_0:def_4; then f . ((s ^\ k) . n) < 0 by A3; hence (f /* (s ^\ k)) . n < 0 by A16, A14, FUNCT_2:108, XBOOLE_1:1; ::_thesis: verum end; then A18: for n being Element of NAT st 0 <= n holds (f /* (s ^\ k)) . n < 0 ; s ^\ k is divergent_to+infty by A11, Th26; then ( f /* (s ^\ k) is convergent & lim (f /* (s ^\ k)) = 0 ) by A1, A2, A17, Def12; then A19: (f /* (s ^\ k)) " is divergent_to-infty by A18, Th36; (f /* (s ^\ k)) " = ((f /* s) ^\ k) " by A12, A15, VALUED_0:27, XBOOLE_1:1 .= ((f /* s) ") ^\ k by SEQM_3:18 .= ((f ^) /* s) ^\ k by A12, RFUNCT_2:12 ; hence (f ^) /* s is divergent_to-infty by A19, Th7; ::_thesis: verum end; theorem :: LIMFUNC1:114 for f being PartFunc of REAL,REAL st f is convergent_in-infty & lim_in-infty f = 0 & ex r being Real st for g being Real st g in (dom f) /\ (left_open_halfline r) holds 0 < f . g holds f ^ is divergent_in-infty_to+infty proof let f be PartFunc of REAL,REAL; ::_thesis: ( f is convergent_in-infty & lim_in-infty f = 0 & ex r being Real st for g being Real st g in (dom f) /\ (left_open_halfline r) holds 0 < f . g implies f ^ is divergent_in-infty_to+infty ) assume that A1: f is convergent_in-infty and A2: lim_in-infty f = 0 ; ::_thesis: ( for r being Real ex g being Real st ( g in (dom f) /\ (left_open_halfline r) & not 0 < f . g ) or f ^ is divergent_in-infty_to+infty ) given r being Real such that A3: for g being Real st g in (dom f) /\ (left_open_halfline r) holds 0 < f . g ; ::_thesis: f ^ is divergent_in-infty_to+infty thus for r1 being Real ex g1 being Real st ( g1 < r1 & g1 in dom (f ^) ) :: according to LIMFUNC1:def_10 ::_thesis: for seq being Real_Sequence st seq is divergent_to-infty & rng seq c= dom (f ^) holds (f ^) /* seq is divergent_to+infty proof let r1 be Real; ::_thesis: ex g1 being Real st ( g1 < r1 & g1 in dom (f ^) ) consider g1 being Real such that A4: g1 < r1 and g1 in dom f by A1, Def9; now__::_thesis:_ex_g2_being_Real_st_ (_g2_<_r1_&_g2_in_dom_(f_^)_) percases ( g1 <= r or r <= g1 ) ; supposeA5: g1 <= r ; ::_thesis: ex g2 being Real st ( g2 < r1 & g2 in dom (f ^) ) consider g2 being Real such that A6: g2 < g1 and A7: g2 in dom f by A1, Def9; take g2 = g2; ::_thesis: ( g2 < r1 & g2 in dom (f ^) ) thus g2 < r1 by A4, A6, XXREAL_0:2; ::_thesis: g2 in dom (f ^) g2 < r by A5, A6, XXREAL_0:2; then g2 in { r2 where r2 is Real : r2 < r } ; then g2 in left_open_halfline r by XXREAL_1:229; then g2 in (dom f) /\ (left_open_halfline r) by A7, XBOOLE_0:def_4; then 0 <> f . g2 by A3; then not f . g2 in {0} by TARSKI:def_1; then not g2 in f " {0} by FUNCT_1:def_7; then g2 in (dom f) \ (f " {0}) by A7, XBOOLE_0:def_5; hence g2 in dom (f ^) by RFUNCT_1:def_2; ::_thesis: verum end; supposeA8: r <= g1 ; ::_thesis: ex g2 being Real st ( g2 < r1 & g2 in dom (f ^) ) consider g2 being Real such that A9: g2 < r and A10: g2 in dom f by A1, Def9; take g2 = g2; ::_thesis: ( g2 < r1 & g2 in dom (f ^) ) g2 < g1 by A8, A9, XXREAL_0:2; hence g2 < r1 by A4, XXREAL_0:2; ::_thesis: g2 in dom (f ^) g2 in { r2 where r2 is Real : r2 < r } by A9; then g2 in left_open_halfline r by XXREAL_1:229; then g2 in (dom f) /\ (left_open_halfline r) by A10, XBOOLE_0:def_4; then 0 <> f . g2 by A3; then not f . g2 in {0} by TARSKI:def_1; then not g2 in f " {0} by FUNCT_1:def_7; then g2 in (dom f) \ (f " {0}) by A10, XBOOLE_0:def_5; hence g2 in dom (f ^) by RFUNCT_1:def_2; ::_thesis: verum end; end; end; hence ex g1 being Real st ( g1 < r1 & g1 in dom (f ^) ) ; ::_thesis: verum end; let s be Real_Sequence; ::_thesis: ( s is divergent_to-infty & rng s c= dom (f ^) implies (f ^) /* s is divergent_to+infty ) assume that A11: s is divergent_to-infty and A12: rng s c= dom (f ^) ; ::_thesis: (f ^) /* s is divergent_to+infty consider k being Element of NAT such that A13: for n being Element of NAT st k <= n holds s . n < r by A11, Def5; A14: rng (s ^\ k) c= rng s by VALUED_0:21; dom (f ^) = (dom f) \ (f " {0}) by RFUNCT_1:def_2; then A15: dom (f ^) c= dom f by XBOOLE_1:36; then A16: rng s c= dom f by A12, XBOOLE_1:1; then A17: rng (s ^\ k) c= dom f by A14, XBOOLE_1:1; now__::_thesis:_for_n_being_Element_of_NAT_holds_0_<_(f_/*_(s_^\_k))_._n let n be Element of NAT ; ::_thesis: 0 < (f /* (s ^\ k)) . n s . (n + k) < r by A13, NAT_1:12; then (s ^\ k) . n < r by NAT_1:def_3; then (s ^\ k) . n in { g2 where g2 is Real : g2 < r } ; then ( (s ^\ k) . n in rng (s ^\ k) & (s ^\ k) . n in left_open_halfline r ) by VALUED_0:28, XXREAL_1:229; then (s ^\ k) . n in (dom f) /\ (left_open_halfline r) by A17, XBOOLE_0:def_4; then 0 < f . ((s ^\ k) . n) by A3; hence 0 < (f /* (s ^\ k)) . n by A16, A14, FUNCT_2:108, XBOOLE_1:1; ::_thesis: verum end; then A18: for n being Element of NAT st 0 <= n holds 0 < (f /* (s ^\ k)) . n ; s ^\ k is divergent_to-infty by A11, Th27; then ( f /* (s ^\ k) is convergent & lim (f /* (s ^\ k)) = 0 ) by A1, A2, A17, Def13; then A19: (f /* (s ^\ k)) " is divergent_to+infty by A18, Th35; (f /* (s ^\ k)) " = ((f /* s) ^\ k) " by A12, A15, VALUED_0:27, XBOOLE_1:1 .= ((f /* s) ") ^\ k by SEQM_3:18 .= ((f ^) /* s) ^\ k by A12, RFUNCT_2:12 ; hence (f ^) /* s is divergent_to+infty by A19, Th7; ::_thesis: verum end; theorem :: LIMFUNC1:115 for f being PartFunc of REAL,REAL st f is convergent_in-infty & lim_in-infty f = 0 & ex r being Real st for g being Real st g in (dom f) /\ (left_open_halfline r) holds f . g < 0 holds f ^ is divergent_in-infty_to-infty proof let f be PartFunc of REAL,REAL; ::_thesis: ( f is convergent_in-infty & lim_in-infty f = 0 & ex r being Real st for g being Real st g in (dom f) /\ (left_open_halfline r) holds f . g < 0 implies f ^ is divergent_in-infty_to-infty ) assume that A1: f is convergent_in-infty and A2: lim_in-infty f = 0 ; ::_thesis: ( for r being Real ex g being Real st ( g in (dom f) /\ (left_open_halfline r) & not f . g < 0 ) or f ^ is divergent_in-infty_to-infty ) given r being Real such that A3: for g being Real st g in (dom f) /\ (left_open_halfline r) holds f . g < 0 ; ::_thesis: f ^ is divergent_in-infty_to-infty thus for r1 being Real ex g1 being Real st ( g1 < r1 & g1 in dom (f ^) ) :: according to LIMFUNC1:def_11 ::_thesis: for seq being Real_Sequence st seq is divergent_to-infty & rng seq c= dom (f ^) holds (f ^) /* seq is divergent_to-infty proof let r1 be Real; ::_thesis: ex g1 being Real st ( g1 < r1 & g1 in dom (f ^) ) consider g1 being Real such that A4: g1 < r1 and g1 in dom f by A1, Def9; now__::_thesis:_ex_g2_being_Real_st_ (_g2_<_r1_&_g2_in_dom_(f_^)_) percases ( g1 <= r or r <= g1 ) ; supposeA5: g1 <= r ; ::_thesis: ex g2 being Real st ( g2 < r1 & g2 in dom (f ^) ) consider g2 being Real such that A6: g2 < g1 and A7: g2 in dom f by A1, Def9; take g2 = g2; ::_thesis: ( g2 < r1 & g2 in dom (f ^) ) thus g2 < r1 by A4, A6, XXREAL_0:2; ::_thesis: g2 in dom (f ^) g2 < r by A5, A6, XXREAL_0:2; then g2 in { r2 where r2 is Real : r2 < r } ; then g2 in left_open_halfline r by XXREAL_1:229; then g2 in (dom f) /\ (left_open_halfline r) by A7, XBOOLE_0:def_4; then 0 <> f . g2 by A3; then not f . g2 in {0} by TARSKI:def_1; then not g2 in f " {0} by FUNCT_1:def_7; then g2 in (dom f) \ (f " {0}) by A7, XBOOLE_0:def_5; hence g2 in dom (f ^) by RFUNCT_1:def_2; ::_thesis: verum end; supposeA8: r <= g1 ; ::_thesis: ex g2 being Real st ( g2 < r1 & g2 in dom (f ^) ) consider g2 being Real such that A9: g2 < r and A10: g2 in dom f by A1, Def9; take g2 = g2; ::_thesis: ( g2 < r1 & g2 in dom (f ^) ) g2 < g1 by A8, A9, XXREAL_0:2; hence g2 < r1 by A4, XXREAL_0:2; ::_thesis: g2 in dom (f ^) g2 in { r2 where r2 is Real : r2 < r } by A9; then g2 in left_open_halfline r by XXREAL_1:229; then g2 in (dom f) /\ (left_open_halfline r) by A10, XBOOLE_0:def_4; then 0 <> f . g2 by A3; then not f . g2 in {0} by TARSKI:def_1; then not g2 in f " {0} by FUNCT_1:def_7; then g2 in (dom f) \ (f " {0}) by A10, XBOOLE_0:def_5; hence g2 in dom (f ^) by RFUNCT_1:def_2; ::_thesis: verum end; end; end; hence ex g1 being Real st ( g1 < r1 & g1 in dom (f ^) ) ; ::_thesis: verum end; let s be Real_Sequence; ::_thesis: ( s is divergent_to-infty & rng s c= dom (f ^) implies (f ^) /* s is divergent_to-infty ) assume that A11: s is divergent_to-infty and A12: rng s c= dom (f ^) ; ::_thesis: (f ^) /* s is divergent_to-infty consider k being Element of NAT such that A13: for n being Element of NAT st k <= n holds s . n < r by A11, Def5; A14: rng (s ^\ k) c= rng s by VALUED_0:21; dom (f ^) = (dom f) \ (f " {0}) by RFUNCT_1:def_2; then A15: dom (f ^) c= dom f by XBOOLE_1:36; then A16: rng s c= dom f by A12, XBOOLE_1:1; then A17: rng (s ^\ k) c= dom f by A14, XBOOLE_1:1; now__::_thesis:_for_n_being_Element_of_NAT_holds_(f_/*_(s_^\_k))_._n_<_0 let n be Element of NAT ; ::_thesis: (f /* (s ^\ k)) . n < 0 s . (n + k) < r by A13, NAT_1:12; then (s ^\ k) . n < r by NAT_1:def_3; then (s ^\ k) . n in { g2 where g2 is Real : g2 < r } ; then ( (s ^\ k) . n in rng (s ^\ k) & (s ^\ k) . n in left_open_halfline r ) by VALUED_0:28, XXREAL_1:229; then (s ^\ k) . n in (dom f) /\ (left_open_halfline r) by A17, XBOOLE_0:def_4; then f . ((s ^\ k) . n) < 0 by A3; hence (f /* (s ^\ k)) . n < 0 by A16, A14, FUNCT_2:108, XBOOLE_1:1; ::_thesis: verum end; then A18: for n being Element of NAT st 0 <= n holds (f /* (s ^\ k)) . n < 0 ; s ^\ k is divergent_to-infty by A11, Th27; then ( f /* (s ^\ k) is convergent & lim (f /* (s ^\ k)) = 0 ) by A1, A2, A17, Def13; then A19: (f /* (s ^\ k)) " is divergent_to-infty by A18, Th36; (f /* (s ^\ k)) " = ((f /* s) ^\ k) " by A12, A15, VALUED_0:27, XBOOLE_1:1 .= ((f /* s) ") ^\ k by SEQM_3:18 .= ((f ^) /* s) ^\ k by A12, RFUNCT_2:12 ; hence (f ^) /* s is divergent_to-infty by A19, Th7; ::_thesis: verum end;