reserve k, m, n, p, K, N for Nat;
reserve i for Integer;
reserve x, y, eps for Real;
reserve seq, seq1, seq2 for Real_Sequence;
reserve sq for FinSequence of REAL;

theorem Th25:
  seq is convergent & lim(seq)=x implies for eps st eps>0 holds ex
  N st for n st n>=N holds seq.n>x-eps
proof
  assume
A1: seq is convergent & lim(seq)=x;
  let eps;
  assume eps>0;
  then consider N such that
A2: for n st n>=N holds |.seq.n-x.|<eps by A1,SEQ_2:def 7;
  take N;
  let n;
  assume n>=N;
  then |.seq.n-x.|<eps by A2;
  then seq.n-x>-eps by SEQ_2:1;
  then (seq.n-x)+x>-eps+x by XREAL_1:6;
  hence thesis;
end;
