Infimum of Set of Integers equals Smallest Element

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 $\R$.

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

Proof

By Set of Integers Bounded Below by Real Number has Smallest Element, $S$ has a smallest element, say $n \in S$.

By Smallest Element is Infimum, $n$ is the infimum of $S$.

$\blacksquare$

Also see

• Supremum of Set of Integers equals Greatest Element
• Infimum of Set of Integers is Integer
• Infimum is Unique