reserve X for ComplexUnitarySpace;
reserve g for Point of X;
reserve seq, seq1, seq2 for sequence of X;
reserve Rseq for Real_Sequence;
reserve Cseq,Cseq1,Cseq2 for Complex_Sequence;
reserve z,z1,z2 for Complex;
reserve r for Real;
reserve k,n,m for Nat;

theorem
  ( for n holds Rseq.n = n-root (||.seq.n.||) ) & Rseq is convergent &
  lim Rseq < 1 implies seq is absolutely_summable
proof
  assume that
A1: for n holds Rseq.n = n-root (||.seq.n.||) and
A2: Rseq is convergent & lim Rseq < 1;
  for n holds ||.seq.||.n >= 0 & Rseq.n = n-root (||.seq.||.n)
  proof
    let n;
    ||.seq.||.n = ||.seq.n.|| by CLVECT_2:def 3;
    hence ||.seq.||.n >= 0 by CSSPACE:44;
    Rseq.n = n-root (||.seq.n.||) by A1;
    hence Rseq.n = n-root (||.seq.||.n) by CLVECT_2:def 3;
  end;
  then ||.seq.|| is summable by A2,SERIES_1:28;
  hence thesis;
end;
