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 Th80:
  seq is convergent implies seq is bounded
proof
  assume seq is convergent;
  then consider g such that
A1: for r st r > 0 ex m st for n st n >= m holds ||.seq.n - g.|| < r by Th9;
  consider m1 be Nat such that
A2: for n st n >= m1 holds ||.seq.n - g.|| < 1 by A1;
A3: now
    take p = ||.g.|| + 1;
    0 + 0 < p by CSSPACE:44,XREAL_1:8;
    hence p > 0;
    let n;
    assume n >= m1;
    then
A4: ||.seq.n - g.|| < 1 by A2;
    ||.seq.n.|| - ||.g.|| <= ||.seq.n - g.|| by CSSPACE:48;
    then ||.seq.n.|| - ||.g.|| < 1 by A4,XXREAL_0:2;
    hence ||.seq.n.|| < p by XREAL_1:19;
  end;
  now
    consider M2 such that
A5: M2 > 0 and
A6: for n st n <= m1 holds ||.seq.n.|| < M2 by Th79;
    consider M1 such that
A7: M1 > 0 and
A8: for n st n >= m1 holds ||.seq.n.|| < M1 by A3;
    take M = M1 + M2;
    thus M > 0 by A7,A5;
    let n;
A9: M > 0 + M2 by A7,XREAL_1:8;
A10: now
      assume n <= m1;
      then ||.seq.n.|| < M2 by A6;
      hence ||.seq.n.|| <= M by A9,XXREAL_0:2;
    end;
A11: M > M1 + 0 by A5,XREAL_1:8;
    now
      assume n >= m1;
      then ||.seq.n.|| < M1 by A8;
      hence ||.seq.n.|| <= M by A11,XXREAL_0:2;
    end;
    hence ||.seq.n.|| <= M by A10;
  end;
  hence thesis;
end;
