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 convergent implies seq is Cauchy
proof
  assume seq is convergent;
  then consider h be Point of X such that
A1: for r st r > 0 ex k st for n st n >= k holds dist((seq.n), h) < r;
  let r;
  assume r > 0;
  then consider m1 be Nat such that
A2: for n st n >= m1 holds dist((seq.n), h) < r/2 by A1,XREAL_1:215;
  take k = m1;
  let n, m;
  assume n >= k & m >= k;
  then dist((seq.n), h) < r/2 & dist((seq.m), h) < r/2 by A2;
  then dist((seq.n), (seq.m)) <= dist((seq.n), h) + dist(h, (seq.m)) & dist((
  seq.n) , h) + dist(h, (seq.m)) < r/2 + r/2 by CSSPACE:51,XREAL_1:8;
  hence thesis by XXREAL_0:2;
end;
