reserve a, b, r, M2 for Real;
reserve Rseq,Rseq1,Rseq2 for Real_Sequence;
reserve k, n, m, m1, m2 for Nat;
reserve X for RealUnitarySpace;
reserve g for Point of X;
reserve seq, seq1, seq2 for sequence of X;

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 being Nat holds ||.seq.||.n >= 0 & Rseq.n = n-root (||.seq.||.n)
  proof
    let n be Nat;
    ||.seq.||.n = ||.seq.n.|| by BHSP_2:def 3;
    hence ||.seq.||.n >= 0 by BHSP_1:28;
    Rseq.n = n-root (||.seq.n.||) by A1;
    hence Rseq.n = n-root (||.seq.||.n) by BHSP_2:def 3;
  end;
  hence thesis by A2,SERIES_1:28;
end;
