# Set of Upper Bounds

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.

• Every $s\in S$ serves as lower bound for $B$. So if $B$ has no lower bounds then $S$ must be empty. – drhab Apr 3 '14 at 21:24
• If you have found one of out answers satisfying you should accept it or at least vote it up. – Vanio Begic Apr 4 '14 at 11:50

Every $s\in S$ has the property that $s\le b$ for every $b\in B$. In other words: every $s\in S$ serves as a lower bound for $B$. If $B$ has no lower bounds then it follows immediately that $S=\emptyset$.