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;

theorem
  (seq is non-decreasing implies seq is bounded_below) & (seq is
  non-increasing implies seq is bounded_above)
proof
  thus seq is non-decreasing implies seq is bounded_below
  proof
    assume
A1: seq is non-decreasing;
    take seq.0-1;
    let n;
    seq.0-1<seq.0-0 & seq.0<=seq.n by A1,SEQM_3:11,XREAL_1:15;
    hence thesis by XXREAL_0:2;
  end;
  assume
A2: seq is non-increasing;
  take seq.0+1;
  let n;
  seq.0+0<seq.0+1 & seq.n<=seq.0 by A2,SEQM_3:12,XREAL_1:8;
  hence thesis by XXREAL_0:2;
end;
