# Exists Integer Below Any Real Number

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## Theorem

Let $x$ be a real number.

Then there exists an integer less than $x$:

- $\forall x \in \R: \exists n \in \Z: n < x$

## Proof

Clearly we may assume without loss of generality that $x < 0$.

From the Archimedean Principle:

- $\exists m \in \N: m > -x$

By Real Numbers form Totally Ordered Field, we have that $\R$ is a totally ordered field.

Therefore by property $(3)$ of Properties of Totally Ordered Field, $\Z \owns -m < x$.

$\blacksquare$