Set of Integers Bounded Below has Smallest Element
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Theorem
Bounded Below by Integer
Let $\Z$ be the set of integers.
Let $\le$ be the ordering on the integers.
Let $\O \subset S \subseteq \Z$ such that $S$ is bounded below in $\struct {\Z, \le}$.
Then $S$ has a smallest element.
Bounded Below by Real Number
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.