
theorem
  for L be ExtREAL_sequence, K be R_eal st K <> +infty & (for n be Nat
  holds L.n <= K) holds sup rng L < +infty
proof
  let L be ExtREAL_sequence, K be R_eal;
  assume that
A1: K <> +infty and
A2: for n be Nat holds L.n <= K;
  now
    let x be ExtReal;
    assume x in rng L;
    then ex z be object st z in dom L & x=L.z by FUNCT_1:def 3;
    hence x <= K by A2;
  end;
  then K is UpperBound of rng L by XXREAL_2:def 1;
  then sup rng L <= K by XXREAL_2:def 3;
  hence thesis by A1,XXREAL_0:2,4;
end;
