Infimum 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 below in the set of real numbers.

Then its infimum $\inf S$ is an integer.

Proof
By Infimum of Set of Integers equals Smallest Element, $S$ has a smallest element $n \in \Z$, that is equals to the infimum of $S$.

Also see

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