
theorem Th813:
  for X be RealBanachSpace, x be sequence of X
    st X is Reflexive & ||.x.|| is bounded holds
     ex x0 be sequence of X st
        x0 is subsequence of x & x0 is weakly-convergent
proof
  let X be RealBanachSpace, x be sequence of X;
  assume that
A2: X is Reflexive and
A3: ||.x.|| is bounded;
  set L = ClNLin(rng x);
  reconsider L0 = the carrier of L as Subset of X by DUALSP01:def 16;
LM1: for z be object st z in rng x holds z in the carrier of L
  proof
    let z be object;
    assume z in rng x; then
C14: z in Lin(rng x) by RLVECT_3:15;
    ex Z be Subset of X st
      Z = the carrier of Lin(rng x)
    & L = NORMSTR(# Cl(Z), Zero_(Cl(Z), X), Add_(Cl(Z), X),
                    Mult_(Cl(Z), X), Norm_(Cl(Z),X) #) by NORMSP_3:def 20;
    hence z in the carrier of L by NORMSP_3:4,C14,TARSKI:def 3;
  end;
  per cases;
   suppose XX1: rng x c= {0.X};
     take x;
     thus x is subsequence of x by VALUED_0:19;
     thus x is weakly-convergent by XX1,WEAKLM1;
   end;
   suppose not rng x c= {0.X}; then
     consider z be object such that
B11:   z in rng x & not z in {0.X};
B12: z in rng x & z <> 0.X by B11,TARSKI:def 1;
B17: z in the carrier of L by LM1,B11;
     0.X = 0.L by DUALSP01:def 16; then
A11: L is non trivial by B12,B17;
A12: L is Reflexive by A2,NORMSP_4:29;
     DualSp DualSp L is separable by A11,A12,NORMSP_4:25; then
A4:  DualSp L is separable by Th73;
     set F=BidualFunc L;
     rng x c= the carrier of L by LM1; then
     reconsider x1 = x as sequence of L by FUNCT_2:6;
     for i be Nat holds ||.x1.||.i = ||.x.||.i
     proof
       let i be Nat;
       thus ||.x1.||.i = ||.x1.i.|| by NORMSP_0:def 4
                 .= ||.x.i.||  by NORMSP_3:28
                 .= ||.x.||.i by NORMSP_0:def 4;
     end; then
B13: ||.x1.|| = ||.x.||; then
     consider r be Real such that
A5:    for n be Nat holds ||.x1.||.n < r by A3,SEQ_2:def 3;
     for n be Nat holds ||.F*x1.||.n < r
     proof
       let n be Nat;
C2:    ||.x1.||.n < r by A5;
       ||.x1.n.|| = ||.F.(x1.n).|| by A11,DUALSP02:9; then
       ||.F.(x1.n).|| < r by C2,NORMSP_0:def 4; then
       ||.(F*x1).n.|| < r by ORDINAL1:def 12,FUNCT_2:15;
       hence thesis by NORMSP_0:def 4;
     end; then
A6:  ||.F*x1.|| is bounded_above;
     consider s be Real such that
A7:    for n be Nat holds s < ||.x1.||.n by B13,A3,SEQ_2:def 4;
     for n be Nat holds s < ||.F*x1.||.n
     proof
       let n be Nat;
C4:    s < ||.x1.||.n by A7;
       ||.x1.n.|| = ||.F.(x1.n).|| by A11,DUALSP02:9; then
       s < ||.F.(x1.n).|| by C4,NORMSP_0:def 4; then
       s < ||.(F*x1).n.|| by ORDINAL1:def 12,FUNCT_2:15;
       hence thesis by NORMSP_0:def 4;
     end; then
     ||.F*x1.|| is bounded_below; then
     consider f1 be sequence of DualSp DualSp L such that
A9:    f1 is subsequence of F*x1
     & f1 is weakly*-convergent by A4,A6,Th814;
     consider L1 be increasing sequence of NAT such that
A91:   f1 = (F*x1)*L1 by A9,VALUED_0:def 17;
     reconsider x01=x1*L1 as sequence of L;
     f1=F*x01 by A91,RELAT_1:36; then
HX:  x01 is weakly-convergent by A9,A12,Lm813A;
BX:  the carrier of L c= the carrier of X by DUALSP01:def 16; then
     reconsider x0=x01 as sequence of X by FUNCT_2:7;
     reconsider y = w-lim x01 as Point of X by BX;
     for h be Lipschitzian linear-Functional of X
       holds h*x0 is convergent & lim (h*x0) = h.y
     proof
       let h be Lipschitzian linear-Functional of X;
       reconsider h1=h| (the carrier of L)
         as Lipschitzian linear-Functional of L by NORMSP_4:22;
GX3:   dom h1 = the carrier of L by FUNCT_2:def 1; then
       rng x01 c= dom h1; then
GX2:   h1*x01 = h*x0 by RELAT_1:165;
       h1*x01 is convergent & lim (h1*x01) = h1.(w-lim x01) by HX,DefWeaklim;
       hence thesis by GX2,GX3,FUNCT_1:47;
     end; then
     x0 is weakly-convergent;
     hence thesis;
  end;
end;
