Set of Integers Bounded Below by Real Number has Smallest Element

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.

Also see

 * Set of Integers Bounded Below has Smallest Element
 * Set of Integers Bounded Above by Real Number has Greatest Element