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 convergent implies for p st 0<p ex n st
  for m,l be Nat st n <= m & n <= l holds |.seq.m-seq.l.|<p
proof
  assume
A1: seq is convergent;
  let p;
  assume 0<p;
  then consider n such that
A2: for m st n <= m holds |.seq.m-seq.n.|<p/2 by A1,Th46;
  take n;
    let m,l be Nat;
    assume n <= m & n <= l;
    then |.seq.m-seq.n.|<p/2 & |.seq.l-seq.n.|<p/2 by A2;
    then
A3: |.seq.m-seq.n.|+|.seq.l-seq.n.| < p/2+p/2 by XREAL_1:8;
    |.(seq.m-seq.n)-(seq.l-seq.n).| <= |.seq.m-seq.n.|+|.seq.l-seq.n.| by
COMPLEX1:57;
    hence |.seq.m-seq.l.| < p by A3,XXREAL_0:2;
end;
