
theorem Th32:
  for X being non empty compact TopSpace
  for seq being sequence of (C_Normed_Algebra_of_ContinuousFunctions X)
    st seq is CCauchy holds seq is convergent
proof
  let X be non empty compact TopSpace;
  let vseq be sequence of (C_Normed_Algebra_of_ContinuousFunctions X);
  assume
A1: vseq is CCauchy;
A2:for x being object st x in CContinuousFunctions X holds
         x in ComplexBoundedFunctions the carrier of X by Lm1;
  then
    CContinuousFunctions X c= ComplexBoundedFunctions the carrier of X;
  then
    rng vseq c= ComplexBoundedFunctions the carrier of X;
  then
    reconsider vseq1 = vseq
        as sequence of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
                                                by FUNCT_2:6;
  now
    let e be Real;
    assume
A3:   e > 0;
    consider k being Nat such that
A4:   for n, m being Nat st n >= k & m >= k holds
    ||.((vseq . n) - (vseq . m)).|| < e by A1,A3,CSSPACE3:8;
    take k;
    now
      let n, m be Nat;
      assume n >= k & m >= k;
      then ||.((vseq . n) - (vseq . m)).|| < e by A4;
      hence ||.((vseq1 . n) - (vseq1 . m)).|| < e by Lm9,Lm3;
    end;
    hence for n, m being Nat st n >= k & m >= k holds
      ||.((vseq1 . n) - (vseq1 . m)).|| < e;
  end; then
A5: vseq1 is CCauchy by CSSPACE3:8;
  then
A6: vseq1 is convergent by CC0SP1:27;
  reconsider Y = CContinuousFunctions X
    as Subset of (C_Normed_Algebra_of_BoundedFunctions the carrier of X)
                                                   by A2,TARSKI:def 3;
A7:rng vseq c= CContinuousFunctions X;
  Y is closed by Th31;
  then
    reconsider tv = lim vseq1
      as Point of (C_Normed_Algebra_of_ContinuousFunctions X) by A7,A5,Th30;
  for e being Real st e > 0 holds ex k being Nat st
    for n being Nat st n >= k holds ||.((vseq . n) - tv).|| < e
  proof
    let e be Real;
    assume e > 0;
    then
      consider k being Nat such that
A8:     for n being Nat st n >= k holds
          ||.((vseq1 . n) - (lim vseq1)).|| < e by A6,CLVECT_1:def 16;
    take k;
    now
      let n be Nat;
      assume n >= k;
      then ||.((vseq1 . n) - (lim vseq1)).|| < e by A8;
      hence ||.((vseq . n) - tv).|| < e by Lm9,Lm3;
    end;
    hence for n being Nat st n >= k holds
      ||.((vseq . n) - tv).|| < e;
  end;
  hence vseq is convergent;
end;
