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
  seq1 is bounded & (seq2 is divergent_to+infty or seq2 is
  divergent_to-infty) implies seq1/"seq2 is convergent & lim(seq1/"seq2)=0
proof
  assume that
A1: seq1 is bounded and
A2: seq2 is divergent_to+infty or seq2 is divergent_to-infty;
  seq2" is convergent & lim(seq2")=0 by A2,Th34;
  hence thesis by A1,SEQ_2:25,26;
end;
