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 Th46:
  seq is convergent iff for p st 0<p ex n st for m st n <= m holds
  |.seq.m-seq.n.|<p
proof
A1: now
    assume
A2: for p st 0<p holds ex n st for m st n <= m holds |.seq.m-seq.n.|< p;
    for p be Real st 0<p holds ex n st for m st n <= m holds |.
    Re seq.m-Re seq.n.|<p
    proof
      let p be Real;
      assume
A3:   0<p;
      consider n such that
A4:   for m st n <= m holds |.seq.m-seq.n.|<p by A2,A3;
      take n;
      let m;
      assume n <= m;
      then
A5:   |.seq.m-seq.n.|<p by A4;
      Re(seq.m-seq.n)=Re(seq.m)-Re(seq.n) by COMPLEX1:19
        .=Re seq.m-Re(seq.n) by Def5
        .=Re seq.m-Re seq.n by Def5;
      then |.Re seq.m-Re seq.n.| <= |.seq.m-seq.n.| by Th13;
      hence thesis by A5,XXREAL_0:2;
    end;
    hence
A6: Re seq is convergent by SEQ_4:41;
    for p be Real st 0<p holds ex n st for m st n <= m holds |.
    Im seq.m-Im seq.n.|<p
    proof
      let p be Real;
      assume
A7:   0<p;
      consider n such that
A8:   for m st n <= m holds |.seq.m-seq.n.|<p by A2,A7;
      take n;
      let m;
      assume n <= m;
      then
A9:   |.seq.m-seq.n.|<p by A8;
      Im(seq.m-seq.n)=Im(seq.m)-Im(seq.n) by COMPLEX1:19
        .=Im seq.m-Im(seq.n) by Def6
        .=Im seq.m-Im seq.n by Def6;
      then |.Im seq.m-Im seq.n.| <= |.seq.m-seq.n.| by Th13;
      hence thesis by A9,XXREAL_0:2;
    end;
    hence Im seq is convergent by SEQ_4:41;
    hence seq is convergent by A6,Th42;
  end;
  now
    assume seq is convergent;
    then consider z being Complex such that
A10: for p being Real st 0<p ex n st for m st n <= m holds |.seq.m-z.|<p;
    let p;
    assume 0< p;
    then consider n such that
A11: for m st n <= m holds |.seq.m-z.|<p/2 by A10;
    take n;
    let m;
    assume n <= m;
    then
A12: |.seq.m-z.|<p/2 by A11;
    |.seq.n-z.|<p/2 by A11;
    then |.seq.m-z.| + |.seq.n-z.| <p/2+p/2 by A12,XREAL_1:8;
    then
A13: |.seq.m-z.| + |.z-seq.n.| <p by COMPLEX1:60;
    |.seq.m-seq.n.| <= |.seq.m-z.| + |.z-seq.n.| by COMPLEX1:63;
    hence |.seq.m-seq.n.|< p by A13,XXREAL_0:2;
  end;
  hence thesis by A1;
end;
