
theorem Th59:
  for L be ExtREAL_sequence st L is without-infty holds sup rng L
  <> +infty iff ex K be Real st 0 <K & for n be Nat holds L.n <= K
proof
  let L be ExtREAL_sequence;
  assume L is without-infty;
  then
A1: -infty < L.1;
A2: dom L = NAT by FUNCT_2:def 1;
  then
A3: L.1 <= sup rng L by FUNCT_1:3,XXREAL_2:4;
A4: now
    assume sup rng L <> +infty;
    then not sup rng L in {-infty,+infty} by A1,A3,TARSKI:def 2;
    then sup rng L in REAL by XBOOLE_0:def 3,XXREAL_0:def 4;
    then reconsider S = sup rng L as Real;
    take K = max(S,1);
    thus 0 < K by XXREAL_0:25;
    let n be Nat;
    n in NAT by ORDINAL1:def 12;
    then
A5: L.n <= sup rng L by A2,FUNCT_1:3,XXREAL_2:4;
    S <= K by XXREAL_0:25;
    hence L.n <= K by A5,XXREAL_0:2;
  end;
  now
    given K being Real such that
    0 < K and
A6: for n be Nat holds L.n <= K;
    now
      let w be ExtReal;
      assume w in rng L;
      then ex v be object st v in dom L & w = L.v by FUNCT_1:def 3;
      hence w <= K by A6;
    end;
    then K is UpperBound of rng L by XXREAL_2:def 1;
    then
A7: sup rng L <= K by XXREAL_2:def 3;
    K in REAL by XREAL_0:def 1;
    hence sup rng L <> +infty by A7,XXREAL_0:9;
  end;
  hence thesis by A4;
end;
