# Set of Integers Bounded Below by Real Number has Smallest Element

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## Theorem

Let $\Z$ be the set of integers.

Let $\le$ be the usual ordering on the real numbers $\R$.

Let $\O \subset S \subseteq \Z$ such that $S$ is bounded below in $\struct {\R, \le}$.

Then $S$ has a smallest element.

## Proof

Let $S$ be bounded below by $x \in \R$.

By the Archimedean Principle, there exists an integer $n \le x$.

Then $S$ is bounded below by $n$.

By Set of Integers Bounded Below by Integer has Smallest Element, $S$ has a smallest element.

$\blacksquare$