theorem Th57:
  seq is bounded implies inferior_realsequence seq is convergent &
  lim inferior_realsequence seq = upper_bound inferior_realsequence seq
proof
  assume seq is bounded;
  then
  inferior_realsequence seq is non-decreasing & inferior_realsequence seq
  is bounded by Th50,Th56;
  hence thesis by Th24;
end;
