
theorem Lm814A2:
  for X be RealNormSpace, f be sequence of DualSp X, x be Point of X
    st ||.f.|| is bounded holds
  ex f0 be sequence of DualSp X, N be increasing sequence of NAT st
    f0 is subsequence of f & ||.f0.|| is bounded
  & f0#x is convergent & f0#x is subsequence of f#x
  & f0 = f*N
proof
  let X be RealNormSpace, f be sequence of DualSp X, x be Point of X;
  assume ||.f.|| is bounded; then
  consider f0 be sequence of DualSp X such that
A1: f0 is subsequence of f & ||.f0.|| is bounded
  & f0#x is convergent & f0#x is subsequence of f#x by Lm814A1;
  take f0;
  ex N be increasing sequence of NAT st f0 = f*N by A1,VALUED_0:def 17;
  hence thesis by A1;
end;
