
theorem Th57:
  for L be ExtREAL_sequence, K be R_eal st (for n be Nat holds L.n
  <= K) holds sup rng L <= K
proof
  let L be ExtREAL_sequence, K be R_eal;
  assume
A1: 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 A1;
  end;
  then K is UpperBound of rng L by XXREAL_2:def 1;
  hence thesis by XXREAL_2:def 3;
end;
