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 $S$.

$\blacksquare$

Also see

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