This is the definition for supremum I found in Principles of Mathematical Analysis, which seems to be incorrect to me (which probably just means I'm wrong):
Suppose $S$ is an ordered set, $E \subset S$, and $E$ is bounded above. Suppose there exists an $\alpha \in S$ with the following properties:
(i) $\alpha$ is an upper bound of $E$.
(ii) If $\gamma < \alpha$ then $\gamma$ is not an upper bound of $E$.
Then $\alpha$ is called the least upper bound of E [that there is at most one such $\alpha$ is clear from (ii)] or the supremum of $E$.
To me, the implication of (ii) isn't clear. I understand that the supremum functions as a kind of "maximum" for irrational numbers (excuse this imprecise intuition), but I don't see how (ii) completes this definition once we've established that $\alpha$ is an upper bound. It seems like we should say instead something like this:
(ii) If $\gamma \in E$ and $\alpha < \gamma$ then $\alpha$ is not an upper bound of $E$.
Best Answer
Nevermind, this actually makes a lot of sense now. I'd word the intuition as: