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 Axiom of Archimedes, 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.
$\blacksquare$