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
  (for n holds rseq1.n = n-root (|.seq.|.n)) & rseq1 is convergent & lim
  rseq1 < 1 implies seq is absolutely_summable
proof
  assume
A1: ( for n holds rseq1.n = n-root (|.seq.|.n))& rseq1 is convergent &
  lim rseq1 < 1;
  for n holds |.seq.|.n >=0 by Lm3;
  hence |.seq.| is summable by A1,SERIES_1:28;
end;
