If an upper bound is a element of a set, does that imply that upper bound is the supremum of the set?

Question

Given a set S, say that x $\in$ S. If we can show that x is an upper bound of S, is the fact that it is a member of S, and an upper bound enough to conclude that:

x = $sup$(S).

This makes sense in my head but I can't seem to find a theorem or proposition for it?

Answer 1

Yes. If $y$ is any upper bound of $S$, then $y\ge x$ since $x\in S$. Hence, $x$ is the least upper bound of $S$.