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 Th27:
  seq is summable implies for eps st eps>0 holds ex K st
  Partial_Sums(seq).K>Sum(seq)-eps
proof
  assume seq is summable;
  then
A1: Partial_Sums(seq) is convergent by SERIES_1:def 2;
  let eps;
  assume eps>0;
  then consider K such that
A2: for k st k>=K holds Partial_Sums(seq).k>lim(Partial_Sums(seq))-eps
  by A1,Th25;
  take K;
  Sum(seq)=lim(Partial_Sums(seq)) by SERIES_1:def 3;
  hence thesis by A2;
end;
