
theorem
  for X be RealNormSpace, x be sequence of X
    st x is convergent holds x is weakly-convergent & w-lim x = lim x
proof
  let X be RealNormSpace, x be sequence of X such that
A2: x is convergent;
  reconsider x0=lim x as Point of X;
A3: for f be Lipschitzian linear-Functional of X
    holds f*x is convergent & lim (f*x) = f.x0
  proof
    let f be Lipschitzian linear-Functional of X;
B1: dom f = the carrier of X by FUNCT_2:def 1;
    consider K be Real such that
B3:   0 <= K & for x be VECTOR of X
        holds |. f.x .| <= K * ||. x .|| by DUALSP01:def 9;
    for x1, x2 be Point of X
      st x1 in the carrier of X & x2 in the carrier of X
    holds |.f/.x1-f/.x2.| <= (K+1)*||.x1-x2.||
    proof
      let x1,x2 be Point of X;
      assume x1 in the carrier of X & x2 in the carrier of X;
C2:   |.f/.x1-f/.x2.| = |.f.(x1-x2).| by HAHNBAN:19;
C3:   |.f.(x1-x2).| <= K * ||. x1-x2 .|| by B3;
      0 + K <= K + 1 by XREAL_1:8; then
      K * ||. x1-x2 .|| <= (K+1) * ||. x1-x2 .|| by XREAL_1:64;
      hence thesis by XXREAL_0:2,C2,C3;
    end; then
    f is_Lipschitzian_on (the carrier of X) by FUNCT_2:def 1,B3; then
    f is_continuous_on (the carrier of X) by NFCONT_1:46; then
B81:f is_continuous_in x0;
B6: rng x c= the carrier of X; then
    f/*x = f*x by B1,FUNCT_2:def 11;
    hence f*x is convergent & lim (f*x) = f.x0 by B81,A2,B1,B6;
  end; then
A4: x is weakly-convergent;
  now let f be Lipschitzian linear-Functional of X;
    f.(w-lim x - x0) = f.(w-lim x) - f.x0 by HAHNBAN:19
                    .= lim (f*x) - f.x0 by A4,DefWeaklim
                    .= lim (f*x) - lim (f*x) by A3;
    hence f.(w-lim x - x0) = 0;
  end; then
  w-lim x - lim x = 0.X by DUALSP02:8;
  hence thesis by A3,RLVECT_1:21;
end;
