# Set of Integers Bounded Below has Smallest Element

## 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.