Let S be a subset of the reals, and suppose that S is bounded above. Let B be the set of upper bounds of S and suppose that B has no lower bound. What do you conclude about S?
I know that it probably has something to do with the Completeness Axiom. I guessed that S is empty, because B basically "pushed" the upper bound of S lower and lower (because B is not bounded below). I tried to do this more formally: If S has an upper bound x where x ≥ y for all y in S, but there is a set B of zs where z gets smaller and smaller, there will always be a z in B such that z < x. So the set S is empty.