
theorem WEAKLM1:
  for X be RealNormSpace, x be sequence of X
    st rng x c= {0.X} holds x is weakly-convergent
proof
  let X be RealNormSpace, x be sequence of X;
  assume AS: rng x c= {0.X};
  reconsider x0=0.X as Point of X;
  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;
A2: for p be Real, n be Nat st 0 < p holds |.(f*x).n - f.x0.| < p
    proof
     let p be Real, n be Nat;
     assume AS1: 0 < p;
C21: x.n in rng x by FUNCT_2:4,ORDINAL1:def 12;
     (f*x).n = f.(x.n) by ORDINAL1:def 12,FUNCT_2:15; then
     (f*x).n = f.(0.X) by TARSKI:def 1,AS,C21;
     hence |.(f*x).n - f.x0.| < p by AS1,COMPLEX1:44;
    end;
A1: for p be Real st 0 < p ex m be Nat st
          for n be Nat st m <= n holds |.(f*x).n - f.x0.| < p
    proof
      let p be Real;
      assume AS1: 0 < p;
      take m = 0;
      thus thesis by AS1,A2;
    end; then
    f*x is convergent;
    hence thesis by A1,SEQ_2:def 7;
  end;
  hence x is weakly-convergent;
end;
