reserve X for RealUnitarySpace,
  x, g, g1, h for Point of X,
  a, p, r, M, M1, M2 for Real,
  seq, seq1, seq2, seq3 for sequence of X,
  Nseq for increasing sequence of NAT,

  k, l, l1, l2, l3, n, m, m1, m2 for Nat;

theorem
  X is complete & seq is Cauchy implies seq is bounded
 proof
  assume X is complete & seq is Cauchy;
   then seq is convergent;
  hence thesis;
 end;
