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 Th22:
  dseq is convergent & lim(dseq)=number_e
proof
A1: now
    let n be Nat;
    thus (dseq^\1).n = dseq.(n+1) by NAT_1:def 3
      .= (1+1/(n+1)) ^ (n+1) by Def4;
  end;
  then
A2: dseq^\1 is convergent by POWER:59;
  hence dseq is convergent by SEQ_4:21;
  for n being Nat
  holds (dseq^\1).n = (1+1/(n+1)) ^ (n+1) by A1;
  then number_e=lim(dseq^\1) by POWER:def 4;
  hence thesis by A2,SEQ_4:22;
end;
