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 Th25:
  seq is non-increasing bounded_below implies lim seq = lower_bound seq
proof
  assume
A1: seq is non-increasing bounded_below;
  then for n holds lim seq <= seq.n by SEQ_4:38;
  then
A2: lim seq <= lower_bound seq by Th10;
  for n holds lower_bound seq <= seq.n by A1,Th8;
  then lower_bound seq <= lim seq by A1,PREPOWER:1;
  hence thesis by A2,XXREAL_0:1;
end;
