reserve X for ComplexUnitarySpace;
reserve x, y, w, g, g1, g2 for Point of X;
reserve z for Complex;
reserve p, q, r, M, M1, M2 for Real;
reserve seq, seq1, seq2, seq3 for sequence of X;
reserve k,n,m for Nat;
reserve Nseq for increasing sequence of NAT;

theorem
  seq is constant implies seq is Cauchy
proof
  assume
A1: seq is constant;
  let r such that
A2: r > 0;
  take k = 0;
  let n, m such that
  n >= k and
  m >= k;
  dist((seq.n), (seq.m)) = dist((seq.n), (seq.n)) by A1,VALUED_0:23
    .= 0 by CSSPACE:50;
  hence thesis by A2;
end;
