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
  (for n holds |. seq.n .| <= rseq.n) & rseq is convergent & lim(rseq)=0
  implies seq is convergent & lim(seq)=0c
proof
  assume that
A1: for n holds |. seq.n .| <= rseq.n and
A2: rseq is convergent & lim(rseq)=0;
A3: now
    let n;
    abs(Re(seq)).n=|.Re(seq).n.| by SEQ_1:12;
    then 0 <= abs(Re(seq)).n by COMPLEX1:46;
    hence C.n <= abs(Re(seq)).n;
  end;
  C.0=0;
  then
A4: lim(C)=0 by SEQ_4:25;
  now
    let n;
    Re(seq).n = Re(seq.n) by Def5;
    then
A5: abs(Re seq).n = |.Re(seq.n).| by SEQ_1:12;
    |.Re(seq.n).| <= |.seq.n.| & |.seq.n.| <= rseq.n by A1,Th13;
    hence abs(Re seq).n <= rseq.n by A5,XXREAL_0:2;
  end;
  then
A6: abs(Re seq) is convergent & lim(abs(Re seq))=0 by A2,A4,A3,SEQ_2:19,20;
  then
A7: Re(seq) is convergent by SEQ_4:15;
A8: now
    let n;
    abs(Im seq).n=|.Im(seq).n.| by SEQ_1:12;
    then 0 <= abs(Im seq).n by COMPLEX1:46;
    hence C.n <= abs(Im seq).n;
  end;
  now
    let n;
A9: Im(seq).n = Im(seq.n) & abs(Im seq).n = |.Im(seq).n.| by Def6,SEQ_1:12;
    |.Im(seq.n).| <= |.seq.n.| & |.seq.n.| <= rseq.n by A1,Th13;
    hence abs(Im seq).n <= rseq.n by A9,XXREAL_0:2;
  end;
  then
A10: abs(Im seq) is convergent & lim(abs(Im seq))=0 by A2,A4,A8,SEQ_2:19,20;
  then
A11: Im(seq) is convergent by SEQ_4:15;
  lim(Im(seq))=0 by A10,SEQ_4:15;
  then
A12: Im lim(seq)=0 by A7,A11,Th42;
  lim(Re(seq))=0 by A6,SEQ_4:15;
  then Re lim(seq)=0 by A7,A11,Th42;
  hence thesis by A7,A11,A12,Lm1,Th42,COMPLEX1:13;
end;
