Intuition behind characterizations of supremum? Can anyone help me understand the intuition behind the characterization of the supremum? I understand supremums and infinmums in the context of the Completeness Property / Supremum Property, which is: Let $S \subset E$.
(1) $b$ is an upper bound of $S$ if $\forall x \in S$, $x \leq b$.
(2) $\beta$ is the supremum of $S$ if for all upper bound $b$ of $S$, $\beta \leq b$.
However, once you throw an epsilon into the definition, ie. Characterization of the supremum, I'm suddenly lost on what a supremum is. The Characterization of the supremum states that $\beta$ is a supremum if: 
$$\forall \epsilon > 0, \exists s_{\epsilon}\text{ s.t. }\beta - \epsilon \leq s_{\epsilon}.$$
What does that mean?!? 
Thanks so much for the help!!
 A: For β to be a least upper bound, everything smaller must fail to be an upper bound (if there were something smaller that is still an upper bound, then β wouldn't be the least upper bound). That is, given any B < β, B must not be an upper bound. B is an upper bound if it's larger than everything in the set: for all s in the set, s < B. So not being an upper bound would be the negation of that. In other words, to fail to be an upper bound, there must be something in the set larger than B. So for every B smaller than β, there must be something in the set larger than B. If B is smaller than β, then we must be able to get to B by subtracting something from β. So the characterization says that no matter how small an amount (ϵ) we subtract from β, we end up with a number (β-ϵ) that is smaller than some member of the set (sϵ), and thus β-ϵ is not an upper bound.
Or, another way of putting it is there is no "space" between β and the set; if you were standing on the number line at β and stretched out your arms, you would hit a member of the set, no matter how short your arms are; there is no ϵ such that you can go ϵ away from β and not reach the set.
