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 $S$.

$\blacksquare$

Also see

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