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 Th58:
  seq is bounded implies superior_realsequence seq is convergent &
  lim superior_realsequence seq = lower_bound superior_realsequence seq
proof
  assume seq is bounded;
  then
  superior_realsequence seq is non-increasing & superior_realsequence seq
  is bounded by Th51,Th56;
  hence thesis by Th25;
end;
