
theorem
  for X be ComplexNormSpace, seq be sequence of X, rseq1 be
Real_Sequence holds (for n be Nat holds rseq1.n = n-root (||.seq.||.
  n)) & rseq1 is convergent & lim rseq1 < 1 implies seq is norm_summable
proof
  let X be ComplexNormSpace;
  let seq be sequence of X;
  let rseq1 be Real_Sequence;
  assume
A1: ( for n be Nat holds rseq1.n = n-root (||.seq.||.n))&
  rseq1 is convergent & lim rseq1 < 1;
  for n be Nat holds ||.seq.||.n >=0 by Th2;
  hence ||.seq.|| is summable by A1,SERIES_1:28;
end;
