Supremum of Set of Integers equals Greatest Element

Theorem

Let $S \subset \Z$ be a non-empty subset of the set of integers.

Let $S$ be bounded above in the set of real numbers $\R$.

Then $S$ has a greatest element, and it is equal to the supremum $\sup S$.

Proof

By Set of Integers Bounded Above by Real Number has Greatest Element, $S$ has a greatest element, say $n \in S$.

By Greatest Element is Supremum, $n$ is the supremum of $S$.

$\blacksquare$

Also see

• Infimum of Set of Integers equals Smallest Element
• Supremum of Set of Integers is Integer
• Supremum is Unique