# Supremum of Set of Integers is Integer

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## Theorem

Let $S \subset \Z$ be a non-empty subset of the set of integers.

Let $S$ be bounded above in the set of real numbers.

Then its supremum $\sup S$ is an integer.

## Proof

By Supremum of Set of Integers equals Greatest Element, $S$ has a greatest element $n \in \Z$, that is equals to the supremum of $S$.

$\blacksquare$