
theorem
  for X be RealNormSpace, g be sequence of DualSp X
    st g is convergent holds g is weakly*-convergent & w*-lim g = lim g
proof
  let X be RealNormSpace, g be sequence of DualSp X such that
A2: g is convergent;
  reconsider g0=lim g as Point of DualSp X;
A3: for x be Point of X
      holds g#x is convergent & lim (g#x) = g0.x
  proof
    let x be Point of X;
B2: for r be Real st 0 < r
      ex m be Nat st
        for n be Nat st m <= n holds |.(g#x).n - g0.x.| < r
    proof
      let r be Real;
      set p = r/ (||.x.|| + 1);
      assume C00: 0 < r; then
      consider m be Nat such that
C1:     for n be Nat st m <= n holds ||.g.n - g0.|| < p
         by A2,NORMSP_1:def 7;
      p*(||.x.|| + 1) = r by XCMPLX_1:87; then
C0:   0 < p & p*||.x.|| < r by C00,XREAL_1:29,68;
CX:   for n be Nat st m <= n holds |.(g#x).n - g0.x.| < r
      proof
        let n be Nat;
        assume m <= n; then
        ||.g.n - g0.|| < p by C1; then
D4:     ||.g.n - g0.|| * ||.x.|| <= p * ||.x.|| by XREAL_1:64;
D2:     |.(g#x).n - g0.x.| = |.(g.n).x - g0.x.| by Def1
                          .= |.(g.n - g0).x.| by DUALSP01:33;
        reconsider h=g.n - g0 as Lipschitzian linear-Functional of X
          by DUALSP01:def 10;
        |.h.x.| <= ||.g.n - g0.|| * ||.x.|| by DUALSP01:26; then
        |.h.x.| <= p * ||.x.|| by D4,XXREAL_0:2;
        hence thesis by D2,C0,XXREAL_0:2;
      end;
      take m;
      thus thesis by CX;
    end; then
    g#x is convergent;
    hence thesis by B2,SEQ_2:def 7;
  end; then
  g is weakly*-convergent;
  hence thesis by A3,Def2;
end;
