Set of Integers Bounded Above by Integer has Greatest Element

Theorem
Let $\Z$ be the set of integers.

Let $\le$ be the ordering on the integers.

Let $\O \subset S \subseteq \Z$ such that $S$ is bounded above in $\struct {\Z, \le}$.

Then $S$ has a greatest element.

Also see

 * Set of Integers Bounded Above has Greatest Element
 * Set of Integers Bounded Below by Integer has Smallest Element
 * Well-Ordering Principle