Supremum of Set of Integers equals Greatest Element

Theorem
Let $S \subset \Z$ be a nonempty subset of the set of integers.

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

Then $S$ has a greatest element, and it is equal to the supremum $\sup S$.

Proof
By Set of Integers Bounded Above by Real Number has Greatest Element, $S$ has a greatest element, say $n\in S$.

By Greatest Element is Supremum, $n$ is the supremum of $S$.

Also see

 * Infimum of Set of Integers equals Smallest Element
 * Supremum of Set of Integers is Integer
 * Supremum is Unique