# Set of Integers Bounded Above has Greatest Element

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## Theorem

### Bounded above by integer

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.

### Bounded above by real number

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.