reserve r,r1,g for Real,
  n,m,k for Nat,
  seq,seq1, seq2 for Real_Sequence,
  f,f1,f2 for PartFunc of REAL,REAL,
  x for set;
reserve r,r1,r2,g,g1,g2 for Real;

theorem
  seq is monotone implies seq is convergent or seq is divergent_to+infty
  or seq is divergent_to-infty
proof
  assume
A1: seq is monotone;
  now
    per cases by A1,SEQM_3:def 5;
    suppose
A2:   seq is non-decreasing;
      now
        per cases;
        suppose
          seq is bounded_above;
          hence thesis by A2;
        end;
        suppose
          not seq is bounded_above;
          hence thesis by A2,Th29;
        end;
      end;
      hence thesis;
    end;
    suppose
A3:   seq is non-increasing;
      now
        per cases;
        suppose
          seq is bounded_below;
          hence thesis by A3;
        end;
        suppose
          not seq is bounded_below;
          hence thesis by A3,Th30;
        end;
      end;
      hence thesis;
    end;
  end;
  hence thesis;
end;
