
theorem Th26:
  for X be RealBanachSpace for seq be sequence of X holds seq is
  norm_summable implies seq is summable
proof
  let X be RealBanachSpace;
  let seq be sequence of X;
  assume seq is norm_summable;
  then
A1: Partial_Sums(||.seq.||) is convergent by SERIES_1:def 2;
  now
    let p be Real;
    assume 0<p;
    then consider n be Nat such that
A2: for m be Nat st n <= m
     holds |.Partial_Sums(||.seq.||).m-Partial_Sums(||.seq.||).n.|< p
    by A1,SEQ_4:41;
     reconsider n as Nat;
    take n;
    let m be Nat;
    assume
A3: n <= m;
    then
A4: ||.Partial_Sums(seq).m-Partial_Sums(seq).n.|| <= |.Partial_Sums(||.
    seq.||).m-Partial_Sums(||.seq.||).n.| by Th25;
    |.Partial_Sums(||.seq.||).m-Partial_Sums(||.seq.||).n.|< p by A2,A3;
    hence ||.Partial_Sums(seq).m-Partial_Sums(seq).n.|| <p by A4,XXREAL_0:2;
  end;
  hence thesis by Th24;
end;
