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
  r>0 & (ex m st for n st n>=m holds |.seq.n.| >= r) implies not |.seq.|
  is convergent or lim |.seq.| <> 0
proof
  assume that
A1: r>0 and
A2: ex m st for n st n>=m holds |.seq.n.| >= r;
  consider m such that
A3: for n st n>=m holds |.seq.n.| >= r by A2;
  now
    let n such that
A4: n>=m;
    0 <= |.seq.|.n by Lm3;
    then |.(|.seq.|).n.| =|.seq.|.n by ABSVALUE:def 1
      .=|.seq.n.| by VALUED_1:18;
    hence |.(|.seq.|).n.| >= r by A3,A4;
  end;
  hence thesis by A1,SERIES_1:38;
end;
