Let S be a nonempty bounded subset of $\Re$ and let $m=supS$. Prove that $m\in S$ iff $m = maxS$.
Here's my proof:
Let $S \subseteq \Re$ be nonempty and let $m=supS$.
Suppose $m \in S $. Since S is bounded, we have S has an upper bound, specifically $m$ which is the least upper bound. Since $m \in S$, that must mean $m = maxS$.
On the other hand, suppose $m = maxS$. This, by definition, implies that $m$ is the least upper bound and $m \in S$.
Q.E.D.
Is this proof correct?