
theorem Lm814A1:
  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 st
      f0 is subsequence of f & ||.f0.|| is bounded
    & f0#x is convergent & f0#x is subsequence of f#x
proof
  let X be RealNormSpace, f be sequence of DualSp X, x be Point of X;
  assume ||.f.|| is bounded; then
  consider r0 be Real such that
B0: 0 < r0 & for m be Nat holds |. ||.f.||.m .| < r0 by SEQ_2:3;
    set r=r0*||.x.|| + 1;
BS: for m be Nat holds |.(f#x).m.| < r
  proof
    let m be Nat;
    reconsider h=f.m as Lipschitzian linear-Functional of X
      by DUALSP01:def 10;
    |. h.x .| = |. (f#x).m .| by Def1; then
B5: |. (f#x).m .| <= ||.f.m.|| * ||.x.|| by DUALSP01:26;
    |. ||.f.||.m .| <= r0 & ||.f.||.m = ||.f.m.|| by B0,NORMSP_0:def 4; then
    ||.f.m.|| <= r0 by ABSVALUE:def 1; then
    ||.f.m.|| * ||.x.|| <= r0 * ||.x.|| & r0*||.x.|| < r by XREAL_1:29,64; then
    ||.f.m.|| * ||.x.|| < r by XXREAL_0:2;
    hence thesis by B5,XXREAL_0:2;
  end;
  reconsider seq=f#x as Real_Sequence;
  consider seq1 be Real_Sequence such that
X1: seq1 is subsequence of seq & seq1 is convergent by B0,SEQ_2:3,BS,SEQ_4:40;
  consider N be increasing sequence of NAT such that
X2: seq1 = seq * N by X1,VALUED_0:def 17;
  reconsider f0=f*N as sequence of DualSp X;
  now let k be Nat;
    thus (f0#x).k = (f0.k).x by Def1
                 .= (f.(N.k)).x by ORDINAL1:def 12,FUNCT_2:15
                 .= (f#x).(N.k) by Def1
                 .= seq1.k by X2,ORDINAL1:def 12,FUNCT_2:15;
  end; then
X5: f0#x = seq1;
  for n be Nat holds |.||.f0.||.n.| < r0
  proof
    let n be Nat;
    ||.f0.||.n = ||.f0.n.|| by NORMSP_0:def 4; then
    ||.f0.||.n = ||. f.(N.n) .|| by ORDINAL1:def 12,FUNCT_2:15; then
    ||.f0.||.n = ||.f.||.(N.n) by NORMSP_0:def 4;
    hence thesis by B0;
  end;
  hence thesis by X1,X5,B0,SEQ_2:3;
end;
