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 Th8:
  aseq(k) is convergent & lim(aseq(k))=1
proof
A1: for eps be Real st 0<eps ex N st for n st N<=n
    holds |.aseq(k).n-1.|<eps
  proof
    let eps be Real;
    assume
A2: eps>0;
    consider N such that
A3: N>k/eps by SEQ_4:3;
    take N;
    let n;
    assume
A4: n>=N;
    then n>(k/eps) by A3,XXREAL_0:2;
    then (k/eps)*eps<n*eps by A2,XREAL_1:68;
    then
A5: k<n*eps by A2,XCMPLX_1:87;
A6: n>0 by A2,A3,A4;
    then |.aseq(k).n-1.| = |.(1-(k/n))-1.| by Th7
      .= |.-(k/n).|
      .= |.k/n.| by COMPLEX1:52
      .= k/n by ABSVALUE:def 1;
    hence thesis by A6,A5,XREAL_1:83;
  end;
  thus aseq(k) is convergent
  by A1;
  hence thesis by A1,SEQ_2:def 7;
end;
