I was reading through the definition of upper/lower bounds in Walter Rudin's Mathematical Analysis:
Suppose $S$ is an ordered set, $E \subset S$. If there exists a $\beta \in S$ such that $x \le \beta$ for every $x \in E$, we say that $E$ is bounded above, and call $\beta$ an upper bound of E.
Wouldn't a corollary of this be that any set can only have one of its upper/lower bounds in itself? It would always be the element $x = \beta$ in $x \le \beta$ (or $x \ge \beta$ for lower bounds)?