Real Analysis Supremum of a Set Proof

[TeenieBopper]

Homework Statement

Let S be a non empty set that is bounded about and β = sup S. Prove that for each ε > 0 there exists a point x in S such that x > β - ε.

Homework Equations

The Attempt at a Solution

I don't really know how to begin this. I know it's true; I'm looking at the problem and I'm like, "Well, duh," but I can't prove it. I know that x ≤ β, and that β - ε ≤ x. Is there more that I have to do?edit: never mind. I'm an idiot. Proof by contradiction makes this really easy.

 

[lippyka]

Suppose that for all x in S, x ≤ β - ε. Then β = inf {x | x ∈ S} ≤ β - ε, which is a contradiction. Thus, there exists an x ∈ S such that x > β - ε.