theorem Th40:
  seq is bounded implies ex seq1 st seq1 is subsequence of seq &
  seq1 is convergent
proof
  assume
A1: seq is bounded;
  consider Nseq such that
A2: seq*Nseq is monotone by Th39;
  take seq1=seq*Nseq;
  thus seq1 is subsequence of seq;
  thus thesis by A1,A2,Th36,SEQM_3:29;
end;
