reserve n,m,k,k1,k2 for Nat;
reserve r,r1,r2,s,t,p for Real;
reserve seq,seq1,seq2 for Real_Sequence;
reserve x,y for set;

theorem Th24:
  seq is non-decreasing bounded_above implies lim seq = upper_bound seq
proof
  assume
A1: seq is non-decreasing bounded_above;
  then for n holds seq.n <= lim seq by SEQ_4:37;
  then
A2: upper_bound seq <= lim seq by Th9;
  for n holds seq.n <= upper_bound seq by A1,Th7;
  then lim seq <= upper_bound seq by A1,PREPOWER:2;
  hence thesis by A2,XXREAL_0:1;
end;
