reserve n,k,k1,m,m1,n1,n2,l for Nat;
reserve r,r1,r2,p,p1,g,g1,g2,s,s1,s2,t for Real;
reserve seq,seq1,seq2 for Real_Sequence;
reserve Nseq for increasing sequence of NAT;
reserve x for set;
reserve X,Y for Subset of REAL;

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;
