Set of Integers Bounded Above by Real Number has Greatest Element

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Theorem

Let $\Z$ be the set of integers.

Let $\leq$ be the usual ordering on the real numbers $\R$.

Let $\varnothing \subset S \subseteq \Z$ such that $S$ is bounded above in $(\R, \leq)$.


Then $S$ has a greatest element.


Proof

Let $S$ be bounded above by $x\in\R$.

By the Archimedean Principle, there exists an integer $n\geq x$.

Then $S$ is bounded above by $n$.

By Set of Integers Bounded Above by Integer has Greatest Element, $S$ has a greatest element.

$\blacksquare$


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