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;

theorem
  rseq is convergent implies for p st 0<p ex n st
   for m,l be Nat st n <= m & n <= l holds |.rseq.m-rseq.l.|<p
proof
  assume
A1: rseq is convergent;
  let p;
  assume 0<p;
  then consider n such that
A2: for m st n <= m holds |.rseq.m-rseq.n.|<p/2 by A1,SEQ_4:41;
  take n;
  now
    let m,l be Nat;
    assume n <= m & n <= l;
    then |.rseq.m-rseq.n.|<p/2 & |.rseq.l-rseq.n.|<p/2 by A2;
    then
A3: |.rseq.m-rseq.n.|+|.rseq.l-rseq.n.| < p/2+p/2 by XREAL_1:8;
    |.rseq.m-rseq.l.| = |.(rseq.m-rseq.n)-(rseq.l-rseq.n).|;
    then |.rseq.m-rseq.l.| <= |.rseq.m-rseq.n.|+|.rseq.l-rseq.n.| by
COMPLEX1:57;
    hence |.rseq.m-rseq.l.| < p by A3,XXREAL_0:2;
  end;
  hence thesis;
end;
