reserve rseq, rseq1, rseq2 for Real_Sequence;
reserve seq, seq1, seq2 for Complex_Sequence;
reserve k, n, n1, n2, m for Nat;
reserve p, r for Real;
reserve z for Complex;
reserve Nseq,Nseq1 for increasing sequence of NAT;

theorem
  seq is bounded implies ex seq1 st seq1 is subsequence of seq & seq1 is
  convergent
proof
  assume seq is bounded;
  then consider r being Real such that
A1: 0 < r and
A2: for n holds |.seq.n.| <r by COMSEQ_2:8;
  now
    let n;
    |.Re seq.n.|=|.Re(seq.n).| & |.Re(seq.n).| <= |.seq.n.| by Def5,Th13;
    hence |.Re seq.n.| < r by A2,XXREAL_0:2;
  end;
  then Re seq is bounded by A1,SEQ_2:3;
  then consider rseq1 such that
A3: rseq1 is subsequence of Re seq and
A4: rseq1 is convergent by SEQ_4:40;
  consider Nseq such that
A5: rseq1=Re seq*Nseq by A3,VALUED_0:def 17;
A6: now
    let n;
  n in NAT by ORDINAL1:def 12;
    then |. (seq * Nseq).n .|=|. seq.(Nseq.n) .| by FUNCT_2:15;
    hence |. (seq * Nseq).n .| < r by A2;
  end;
  now
    let n;
    |.Im (seq*Nseq).n.|=|.Im((seq* Nseq).n).| & |.Im((seq* Nseq).n).|
    <= |.( seq* Nseq).n.| by Def6,Th13;
    hence |.Im (seq*Nseq).n.| < r by A6,XXREAL_0:2;
  end;
  then Im (seq*Nseq) is bounded by A1,SEQ_2:3;
  then consider rseq2 such that
A7: rseq2 is subsequence of Im (seq*Nseq) and
A8: rseq2 is convergent by SEQ_4:40;
  consider Nseq1 such that
A9: rseq2=Im (seq*Nseq)*Nseq1 by A7,VALUED_0:def 17;
  Re ((seq*Nseq)*Nseq1)=(Re (seq*Nseq))*Nseq1 by Th22
    .=rseq1*Nseq1 by A5,Th22;
  then
A10: Re ((seq*Nseq)*Nseq1) is convergent by A4,SEQ_4:16;
  seq*Nseq is subsequence of seq & (seq*Nseq)*Nseq1 is subsequence of seq
  *Nseq by VALUED_0:def 17;
  then
A11: (seq*Nseq)*Nseq1 is subsequence of seq by VALUED_0:20;
  Im ((seq*Nseq)*Nseq1) is convergent by A8,A9,Th22;
  hence thesis by A11,A10,Th42;
end;
