reserve X for ComplexUnitarySpace;
reserve g for Point of X;
reserve seq, seq1, seq2 for sequence of X;
reserve Rseq for Real_Sequence;
reserve Cseq,Cseq1,Cseq2 for Complex_Sequence;
reserve z,z1,z2 for Complex;
reserve r for Real;
reserve k,n,m for Nat;

theorem
  for X being ComplexHilbertSpace, seq being sequence of X
  holds seq is Cauchy & Cseq is Cauchy implies Cseq *
  seq is Cauchy
proof
 let X be ComplexHilbertSpace, seq be sequence of X;
  assume that
A1: seq is Cauchy and
A2: Cseq is Cauchy;
  now
    let r be Real;
    assume r > 0;
    then consider k such that
A3: for n, m st n >= k & m >= k holds |.((Cseq.n - Cseq.m)).| < r by A2;
    take k;
    thus for n st n >= k holds |.((Cseq.n - Cseq.k)).| < r by A3;
  end;
  then
A4: Cseq is convergent by COMSEQ_3:46;
   seq is convergent by A1,CLVECT_2:def 11;
  hence thesis by A4,Th48,CLVECT_2:65;
end;
