reserve rseq, rseq1, rseq2 for Real_Sequence;
reserve seq, seq1, seq2 for Complex_Sequence;
reserve k, n, n1, n2, m for Nat;
reserve p, r for Real;
reserve z for Complex;
reserve Nseq,Nseq1 for increasing sequence of NAT;

theorem Th51:
  (for n holds seq.n = 0c) implies seq is absolutely_summable
proof
  assume
A1: for n holds seq.n = 0c;
  take 0;
  let p be Real such that
A2: 0<p;
  take 0;
  let m;
  |.(Partial_Sums |.seq.|).m-0.| = |.0-0.| by A1,Th25
    .= 0 by ABSVALUE:def 1;
  hence thesis by A2;
end;
