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 bounded
proof
  assume seq is constant;
  then consider x such that
A1: for n being Nat holds seq.n = x by VALUED_0:def 18;
A2: x = 09(X) implies seq is bounded
  proof
    consider M being Real such that
A3: M > 0 by XREAL_1:1;
    assume
A4: x = 09(X);
    for n holds ||.seq.n.|| <= M by A3,CSSPACE:42,A1,A4;
    hence thesis by A3;
  end;
  x <> 09(X) implies seq is bounded
  proof
    assume x <> 09(X);
    consider M being Real such that
A5: ||.x.|| < M by XREAL_1:1;
    reconsider M as Real;
    ||.x.|| >= 0 & for n holds ||.seq.n.|| <= M by A1,A5,CSSPACE:44;
    hence thesis by A5;
  end;
  hence thesis by A2;
end;
