theorem Th117:
  S is convergent implies ||.S.|| is convergent
proof
  assume S is convergent;
  then consider g such that
A1: for r st 0 < r ex m st for n st m <= n holds ||.(S.n) - g.|| < r;
  now
    let r be Real;
    assume
A2: 0 < r;
    consider m1 such that
A3: for n st m1 <= n holds ||.(S.n) - g.|| < r by A1,A2;
    take k = m1;
      let n;
      assume k <= n;
      then
A4:   ||.(S.n) - g.|| < r by A3;
      |.||.(S.n).|| - ||.g.||.| <= ||.(S.n) - g.|| by Th110;
      then |.||.(S.n).|| - ||.g.||.| < r by A4,XXREAL_0:2;
      hence |.||.S.||.n - ||.g.||.| < r by NORMSP_0:def 4;
  end;
  hence thesis by SEQ_2:def 6;
end;
