
theorem Lm814A:
  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
proof
  let X be RealNormSpace, f be sequence of DualSp X, x be Point of X;
  assume AS0:||.f.|| is bounded;
  consider r0 be Real such that
B0: 0 < r0 & for m be Nat holds |. ||.f.||.m .| < r0 by AS0,SEQ_2:3;
    set r = r0*||.x.|| + 1;
C1: r0*||.x.|| < r0*||.x.|| + 1 by XREAL_1:29;
BY: 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;
B5: |.h.x.| <= ||.f.m.|| * ||.x.|| by DUALSP01:26;
B3: |. ||.f.||.m .| <= r0 by B0;
    ||.f.||.m = ||.f.m.|| by NORMSP_0:def 4; then
    ||.f.m.|| <= r0 by B3,ABSVALUE:def 1; then
C6: ||.f.m.|| * ||.x.|| <= r0 * ||.x.|| by XREAL_1:64;
    |.(f#x).m .| = |.(f.m).x.| by Def1; then
    |.(f#x).m .| <= r0 * ||.x.|| by B5,XXREAL_0:2,C6;
    hence thesis by C1,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,BY,SEQ_4:40;
  consider N be increasing sequence of NAT such that
X2: seq1 = seq * N by X1,VALUED_0:def 17;
  set f0=f*N;
  for k be Nat holds (f0#x).k = seq1.k
  proof
    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
Y5: f0#x = seq1;
  for n be Nat holds |.||.f0.||.n.| < r0
  proof
    let n be Nat;
Y2: f0.n = f.(N.n) by ORDINAL1:def 12,FUNCT_2:15;
    ||.f0.||.n = ||.f0.n.|| by NORMSP_0:def 4; then
    ||.f0.||.n = ||.f.||.(N.n) by Y2,NORMSP_0:def 4;
    hence thesis by B0;
  end;
  hence thesis by X1,Y5,B0,SEQ_2:3;
end;
