
theorem Th52:
  for seq be ExtREAL_sequence, r be Real st (for n be Nat
  holds seq.n = r) holds seq is convergent_to_finite_number & lim seq = r
proof
  let seq be ExtREAL_sequence;
  let r be Real;
  assume
A1: for n be Nat holds seq.n = r;
A2: now
    reconsider n=1 as Nat;
    let p be Real;
    assume
A3: 0 < p;
    take n;
    let m be Nat such that
    n <= m;
    seq.m = r by A1;
    then seq.m - r = 0 by XXREAL_3:7;
    then |. seq.m - r.| = 0 by EXTREAL1:16;
    hence |. seq.m - r.| < p by A3;
  end;
  hence
A4: seq is convergent_to_finite_number;
  then
A5: seq is convergent;
  reconsider r as R_eal by XXREAL_0:def 1;
( ex g be Real st r = g & (for p
be Real st 0<p ex n be Nat st for m be Nat st n<=m holds |.seq.m-r.| <
  p) & seq is convergent_to_finite_number ) by A2,A4;
 hence thesis by Def12,A5;
end;
