Supremum of Set of Integers is Integer

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 $S$.

Also see

 * Infimum of Set of Integers is Integer
 * Supremum is Unique