
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)) & (ex m be Nat st for n be Nat st m<=n holds rseq1.n
  >= 1) implies ||.seq.|| is not 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))& ex
  m be Nat st for n be Nat st m<=n holds rseq1.n>= 1;
  for n be Nat holds ||.seq.||.n >=0 by Th2;
  hence thesis by A1,SERIES_1:29;
end;
