
theorem Th79:
  for X be RealNormSpace, x be sequence of X
     st X is non trivial & x is weakly-convergent
  holds ||.x.|| is bounded & ||. w-lim x .|| <= lim_inf ||.x.||
      & w-lim x in ClNLin(rng x)
proof
  let X be RealNormSpace, x be sequence of X;
  assume AS: X is non trivial & x is weakly-convergent;
  reconsider x0=w-lim x as Point of X;
  for f be Point of DualSp X holds
    ex Kf be Real st 0 <= Kf
    & for y be Point of X st y in rng x holds |. f.y .| <= Kf
  proof
    let f be Point of DualSp X;
    reconsider h=f as Lipschitzian linear-Functional of X
      by DUALSP01:def 10;
    h*x is convergent by AS; then
    consider Kf be Real such that
A1:   0 < Kf
    & for n be Nat holds |. (h*x).n .| < Kf by SEQ_2:3;
    for y be Point of X st y in rng x holds |. f.y .| <= Kf
    proof
      let y be Point of X;
      assume y in rng x; then
      consider m be Nat such that
A2:     y = x.m by NFCONT_1:6;
      |. (h*x).m .| = |. f.(x.m) .| by FUNCT_2:15,ORDINAL1:def 12;
      hence thesis by A2,A1;
    end;
    hence thesis by A1;
  end; then
  consider K be Real such that
A4: 0 <= K
  & for y be Point of X st
      y in rng x holds ||. y .|| <= K by AS,DUALSP02:19;
A5: for n be Nat holds ||.x.||.n <= K
  proof
    let n be Nat;
    n in NAT by ORDINAL1:def 12; then
    ||. x.n .|| <= K by A4,FUNCT_2:4;
    hence thesis by NORMSP_0:def 4;
  end;
A6: K + 0 < K + 1 by XREAL_1:8;
X10: for n be Nat holds |. ||.x.||.n .| < (K+1)
  proof
    let n be Nat;
    ||.x.||.n <= K by A5; then
A7: ||.x.||.n < (K+1) by A6,XXREAL_0:2;
    0 <= ||. x.n .||; then
    0 <= ||.x.||.n by NORMSP_0:def 4;
    hence thesis by A7,ABSVALUE:def 1;
  end; then
X1: ||.x.|| is bounded by A4,SEQ_2:3;
B0: for f be Point of DualSp X holds
    |. f.x0 .| <= (lim_inf ||.x.||) * ||.f.||
  proof
    let f be Point of DualSp X;
    reconsider h=f as Lipschitzian linear-Functional of X
      by DUALSP01:def 10;
B1: h*x is convergent & lim (h*x) = h.x0 by DefWeaklim,AS;
B6: for n be Nat holds |. (h*x) .|.n <= (||.f.||(#)||.x.||).n
    proof
      let n be Nat;
D21:  |. h.(x.n) .| <= ||.f.|| * ||. x.n .|| by DUALSP01:26;
      |. h.(x.n) .| = |. (h*x).n .| by FUNCT_2:15,ORDINAL1:def 12; then
      |. (h*x).n .| <= ||.f.|| * (||.x.||.n) by D21,NORMSP_0:def 4; then
      |.h*x.|.n <= ||.f.|| * (||.x.||.n) by SEQ_1:12;
      hence thesis by SEQ_1:9;
    end;
    ||.f.||(#)||.x.|| is bounded by A4,SEQ_2:3,X10,SEQM_3:37; then
    lim_inf |. (h*x) .| <= lim_inf (||.f.||(#)||.x.||)
      by B1,B6,RINFSUP1:91; then
B7: lim |. (h*x) .| <= lim_inf (||.f.||(#)||.x.||) by B1,RINFSUP1:89;
    lim_inf (||.f.||(#)||.x.||) = (lim_inf ||.x.||) * ||.f.||
      by X1,LOPBAN_5:1;
    hence thesis by SEQ_4:14,B1,B7;
  end;
  now let s be Real;
    assume D5: 0 < s;
    for k be Nat holds 0 - s < ||.x.||.(0+k)
    proof
      let k be Nat;
      ||. x.k .|| = ||.x.||.k by NORMSP_0:def 4;
      hence thesis by D5;
    end;
    hence ex n be Nat st for k be Nat holds 0 - s < ||.x.||.(n+k);
  end; then
B8: 0 <= lim_inf ||.x.|| by X1,RINFSUP1:82;
  consider Y be non empty Subset of REAL such that
B9: Y = {|.(Bound2Lipschitz(F,X)).x0.|
         where F is Point of DualSp X :||.F.|| <= 1 }
  & ||. x0 .|| = upper_bound Y by AS,DUALSP02:7;
X21: now let r be Real;
    assume r in Y; then
    consider F be Point of DualSp X such that
D7:   r = |.(Bound2Lipschitz(F,X)).x0.| & ||.F.|| <= 1 by B9;
    reconsider f1=F as Lipschitzian linear-Functional of X
      by DUALSP01:def 10;
D8: f1 = Bound2Lipschitz(F,X) by DUALSP01:23;
D9: |. F.x0 .| <= (lim_inf ||.x.||) * ||.F.|| by B0;
    (lim_inf ||.x.||) * ||.F.|| <= (lim_inf ||.x.||) * 1
      by B8,D7,XREAL_1:64;
    hence r <= lim_inf ||.x.|| by D7,D8,D9,XXREAL_0:2;
  end;
  x0 in ClNLin(rng x)
  proof
    set M = ClNLin(rng x);
    consider Z be Subset of X such that
C1:   Z = the carrier of Lin(rng x)
    & M = NORMSTR(# Cl(Z), Zero_(Cl(Z),X), Add_(Cl(Z),X), Mult_(Cl(Z),X),
                    Norm_(Cl(Z), X) #) by NORMSP_3:def 20;
    reconsider Y = the carrier of M as Subset of X by DUALSP01:def 16;
C3: Y is linearly-closed by NORMSP_3:29;
    x0 in Y
    proof
      assume AS0: not x0 in Y;
      consider G be Point of DualSp X such that
C5:     (for y be Point of X
           st y in Y holds (Bound2Lipschitz(G,X)).y = 0) and
C6:     (Bound2Lipschitz(G,X)).x0 = 1 by C1,C3,AS0,DUALSP02:2;
      reconsider g=G as Lipschitzian linear-Functional of X
        by DUALSP01:def 10;
C7:   g = Bound2Lipschitz(G,X) by DUALSP01:23;
C8:   g*x is convergent by AS;
C101: for n be Nat holds (g*x).n <= (seq_const 0).n
      proof
        let n be Nat;
        n in NAT by ORDINAL1:def 12; then
        x.n in Lin(rng x) by RLVECT_3:15,FUNCT_2:4;then
        x.n in Y by C1,NORMSP_3:4,TARSKI:def 3; then
        g.(x.n) = 0 by C5,C7; then
        (g*x).n = 0 by FUNCT_2:15,ORDINAL1:def 12;
        hence thesis;
      end;
C111: lim (seq_const 0) = (seq_const 0).0 by SEQ_4:26
                       .= 0;
      lim (g*x) = g.x0 by DefWeaklim,AS
               .= 1 by C6,DUALSP01:23;
      hence contradiction by C111,C101,C8,SEQ_2:18;
    end;
    hence x0 in M;
  end;
  hence thesis by A4,SEQ_2:3,X10,B9,SEQ_4:45,X21;
end;
