:: 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;