Infimum of Set of Integers equals Smallest Element

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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 $\sup S$.


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 $S$.


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