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 Th30:
  seq is non-increasing & not seq is bounded_below implies seq is
  divergent_to-infty
proof
  assume that
A1: seq is non-increasing and
A2: not seq is bounded_below;
  let r;
  consider n such that
A3: seq.n<=r-1 by A2;
  take n;
  let m;
  assume n<=m;
  then seq.m<=seq.n by A1,SEQM_3:8;
  then seq.m<=r-1 by A3,XXREAL_0:2;
  hence thesis by Lm1;
end;
