Set of Integers Bounded Above by Real Number has Greatest Element
Let $\Z$ be the set of integers.
Let $\leq$ be the usual ordering on the real numbers $\R$.
Let $\varnothing \subset S \subseteq \Z$ such that $S$ is bounded above in $(\R, \leq)$.
Then $S$ has a greatest element.
Let $S$ be bounded above by $x\in\R$.
Then $S$ is bounded above by $n$.