reserve r, s, t for Real;
reserve seq for Real_Sequence,
  X, Y for Subset of REAL;

theorem Th39: ::: SEQ_2
  rng seq is real-bounded iff seq is bounded
proof
  thus rng seq is real-bounded implies seq is bounded
  proof
    given s such that
A1: s is LowerBound of rng seq;
    given t such that
A2: t is UpperBound of rng seq;
    thus seq is bounded_above
    proof
      take t+1;
      let n be Nat;
A3:   n in NAT by ORDINAL1:def 12;
      seq.n <= t & t < t+1 by A2,FUNCT_2:4,XREAL_1:29,XXREAL_2:def 1,A3;
      hence thesis by XXREAL_0:2;
    end;
    take s-1;
    let n be Nat;
    s < s+1 by XREAL_1:29;
    then
A4: s-1 < s by XREAL_1:19;
A5:   n in NAT by ORDINAL1:def 12;
    seq.n >= s by A1,FUNCT_2:4,XXREAL_2:def 2,A5;
    hence thesis by A4,XXREAL_0:2;
  end;
  given t such that
A6: for n being Nat holds seq.n<t;
  given s such that
A7: for n being Nat holds seq.n>s;
  thus rng seq is bounded_below
  proof
    take s;
    let r be ExtReal;
    assume r in rng seq;
    then ex n being object st n in dom seq & seq.n = r by FUNCT_1:def 3;
    hence thesis by A7;
  end;
  take t;
  let r be ExtReal;
  assume r in rng seq;
  then ex n being object st n in dom seq & seq.n = r by FUNCT_1:def 3;
  hence thesis by A6;
end;
