
theorem
  for X be RealNormSpace, f be sequence of DualSp X, x be sequence of X
    st ||.f.|| is bounded holds
  ex F be sequence of Funcs(NAT,the carrier of DualSp X) st
     F.0 is subsequence of f & (F.0)#(x.0) is convergent
   & ( for k be Nat holds F.(k+1) is subsequence of F.k )
   & ( for k be Nat holds (F.(k+1))#(x.(k+1)) is convergent)
proof
  let X be RealNormSpace, f be sequence of DualSp X, x be sequence of X;
  assume AS: ||.f.|| is bounded;
  set D = Funcs(NAT,the carrier of DualSp X);
  consider f0 be sequence of DualSp X such that
P0: f0 is subsequence of f & ||.f0.|| is bounded
  & f0#(x.0) is convergent by AS,Lm814A;
  reconsider A = f0 as Element of D by FUNCT_2:8;
  defpred P[Nat,sequence of DualSp X,sequence of DualSp X] means
    ||.$2.|| is bounded implies
       ( $3 is subsequence of $2 & ||.$3.|| is bounded
       & ($3)#(x.($1+1)) is convergent);
P1: for n being Nat for z being Element of D
      ex y being Element of D st P[n,z,y]
  proof
    let n be Nat;
    let z be Element of D;
    consider f0 be sequence of DualSp X such that
X1: ||.z.|| is bounded implies
          f0 is subsequence of z & ||.f0.|| is bounded
        & f0#(x.(n+1)) is convergent by Lm814A;
    reconsider y = f0 as Element of D by FUNCT_2:8;
    take y;
    thus thesis by X1;
  end;
  consider F be sequence of D such that
X2: F.0 = A &
    for n being Nat holds P[n,F.n,F.(n+1)] from RECDEF_1:sch 2(P1);
  defpred Q[Nat] means
    F.($1+1) is subsequence of F.$1 & ||.F.($1+1).|| is bounded
    & (F.($1+1))#(x.($1+1)) is convergent;
Q0: Q[0] by X2,P0;
Q1: for n be Nat st Q[n] holds Q[n+1] by X2;
Q2: for n be Nat holds Q[n] from NAT_1:sch 2(Q0,Q1);
  take F;
  thus thesis by P0,X2,Q2;
end;
