Let A and B be bounded sets of real numbers such that a <= b for every a in A and every b in B. Show that sup?

The proof of this is quite long, which I do not remember. You are trying to show that alpha = sup(A) and beta = inf(B), such that alpha <= beta. For a sup to exist, the set A must be non-empty, a set of real numbers and bounded above. This guarantees a least upper bound exists (See the least upper bound property)