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 Th56:
  seq is bounded implies superior_realsequence seq is bounded &
  inferior_realsequence seq is bounded
proof
  assume
A1: seq is bounded;
  then inferior_realsequence seq is non-decreasing by Th50;
  then
A2: inferior_realsequence seq is bounded_below by LIMFUNC1:1;
  now
    let m;
    upper_bound(inferior_realsequence seq) <= (superior_realsequence seq).m
    by A1,Th54;
    hence (upper_bound(inferior_realsequence seq)) - 1 <
    (superior_realsequence seq).m
    by Lm1;
  end;
  then
A3: (superior_realsequence seq) is bounded_below by SEQ_2:def 4;
  now
    let m;
    (inferior_realsequence seq).m <= lower_bound (superior_realsequence seq)
    by A1,Th53;
    hence (inferior_realsequence seq).m <
    (lower_bound (superior_realsequence seq)) +1
    by Lm1;
  end;
  then
A4: (inferior_realsequence seq) is bounded_above by SEQ_2:def 3;
  superior_realsequence seq is non-increasing by A1,Th51;
  then superior_realsequence seq is bounded_above by LIMFUNC1:1;
  hence thesis by A2,A4,A3;
end;
