
theorem Th55:
  for L,G be ExtREAL_sequence st (for n be Nat holds L.n <= G.n)
  holds sup rng L <= sup rng G
proof
  let L,G be ExtREAL_sequence;
  assume
A1: for n be Nat holds L.n <= G.n;
A2: now
    let n be Element of NAT;
    dom G = NAT by FUNCT_2:def 1;
    then
A3: G.n in rng G by FUNCT_1:def 3;
A4: L.n <= G.n by A1;
    sup rng G is UpperBound of rng G by XXREAL_2:def 3;
    then G.n <= sup rng G by A3,XXREAL_2:def 1;
    hence L.n <= sup rng G by A4,XXREAL_0:2;
  end;
  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 <= sup rng G by A2;
  end;
  then sup rng G is UpperBound of rng L by XXREAL_2:def 1;
  hence thesis by XXREAL_2:def 3;
end;
