reserve n,m,k,k1,k2 for Nat;
reserve X for non empty Subset of ExtREAL;
reserve Y for non empty Subset of REAL;
reserve seq for ExtREAL_sequence;
reserve e1,e2 for ExtReal;
reserve rseq for Real_Sequence;

theorem Th19:
  seq is convergent_to_finite_number implies ex k st seq^\k is bounded
proof
  assume
A1: seq is convergent_to_finite_number;
  then
A2: not (lim seq = +infty & seq is convergent_to_+infty) by MESFUNC5:50;
A3: not (lim seq = -infty & seq is convergent_to_-infty) by A1,MESFUNC5:51;
  seq is convergent by A1,MESFUNC5:def 11;
  then
A4: ex g be Real st lim seq =g & (for p be Real st 0<p ex n be
  Nat st for m be Nat st n<=m holds |.seq.m-lim seq .| < p) & seq is
  convergent_to_finite_number by A2,A3,MESFUNC5:def 12;
  then consider g be Real such that
A5: lim seq = g;
   reconsider jj=1 as Real;
  set UB = g+jj;
  consider k be Nat such that
A6: for m be Nat st k<=m holds |.seq.m-lim seq .| < 1 by A4;
  reconsider K = k as Element of NAT by ORDINAL1:def 12;
  take K;
  now
    let r be ExtReal;
    assume r in rng (seq^\K);
    then consider n be object such that
A7: n in dom(seq^\K) and
A8: r=(seq^\K).n by FUNCT_1:def 3;
    reconsider n as Element of NAT by A7;
    |.seq.(n+k)-lim seq .| <= 1. by A6,NAT_1:11;
    then seq.(n+k)-lim seq <= 1. by EXTREAL1:23;
    then seq.(n+k)+-lim seq +lim seq <= 1. +lim seq by XXREAL_3:36;
    then seq.(n+k)+(-lim seq +lim seq) <= 1. +lim seq by A5,XXREAL_3:29;
    then seq.(n+k)+0. <= 1. +lim seq by XXREAL_3:7;
    then seq.(n+k) <= 1. +lim seq by XXREAL_3:4;
    then seq.(n+k) <= UB by A5,SUPINF_2:1;
    hence r <= UB by A8,NAT_1:def 3;
  end;
  then UB is UpperBound of rng(seq^\K) by XXREAL_2:def 1;
  hence rng (seq^\K) is bounded_above by XXREAL_2:def 10;
  set UL = g-1;
  now
    let r be ExtReal;
    assume r in rng (seq^\K);
    then consider n be object such that
A9: n in dom (seq^\K) and
A10: r=(seq^\K).n by FUNCT_1:def 3;
    reconsider n as Element of NAT by A9;
    |.seq.(n+k)-lim seq .| < 1 by A6,NAT_1:11;
    then -1. <=seq.(n+k)-lim seq by EXTREAL1:23;
    then -1. +lim seq <= seq.(n+k)+-lim seq +lim seq by XXREAL_3:36;
    then -1. +lim seq <= seq.(n+k)+(-lim seq +lim seq) by A5,XXREAL_3:29;
    then
A11: -1. +lim seq <= seq.(n+k)+0. by XXREAL_3:7;
    -1.= -jj by SUPINF_2:2;
    then -jj+ g = -1.+ lim seq by A5,SUPINF_2:1;
    then UL <= seq.(n+k) by A11,XXREAL_3:4;
    hence UL <= r by A10,NAT_1:def 3;
  end;
  then UL is LowerBound of rng (seq^\K) by XXREAL_2:def 2;
  hence rng (seq^\K) is bounded_below by XXREAL_2:def 9;
end;
