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 Th15:
  seq=rseq & seq is convergent_to_finite_number implies rseq is
  convergent & lim seq = lim rseq
proof
  assume that
A1: seq=rseq and
A2: seq is convergent_to_finite_number;
A3: not (lim seq = +infty & seq is convergent_to_+infty) by A2,MESFUNC5:50;
A4: not (lim seq = -infty & seq is convergent_to_-infty) by A2,MESFUNC5:51;
  seq is convergent by A2,MESFUNC5:def 11;
  then consider g be Real such that
A5: lim seq = g and
A6: 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 and
  seq is convergent_to_finite_number by A3,A4,MESFUNC5:def 12;
A7: for p be Real st 0<p
    ex n st for m st n<=m holds |.rseq.m-g qua Complex.| < p
  proof
    let p be Real;
    assume 0 < p;
    then consider n be Nat such that
A8: for m be Nat st n <= m holds |.seq.m-lim seq .| < p by A6;
    reconsider N=n as Element of NAT by ORDINAL1:def 12;
    take N;
    hereby
      let m be Nat;
      assume N <= m;
      then
A9:   |.seq.m-lim seq .| < p by A8;
      rseq.m - g = seq.m - lim seq by A1,A5,SUPINF_2:3;
      hence |.rseq.m - g qua Complex.| < p by A9,EXTREAL1:12;
    end;
  end;
  then rseq is convergent by SEQ_2:def 6;
  hence thesis by A5,A7,SEQ_2:def 7;
end;
