Set of Integers Bounded Above has Greatest Element

Theorem

Bounded above 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 above in $\struct {\Z, \le}$.

Then $S$ has a greatest element.

Bounded above 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 above in $\struct {\R, \le}$.

Then $S$ has a greatest element.

Also see

• Set of Integers Bounded Below has Smallest Element
• Definition:Floor Function