# Real Number is Integer iff equals Floor

## Theorem

Let $x \in \R$.

Then:

$x = \floor x \iff x \in \Z$

where $\floor x$ is the floor of $x$.

## Proof

Let $x = \floor x$.

As $\floor x \in \Z$, then so must $x$ be.

Now let $x \in \Z$.

We have:

$\floor x = \sup \set {m \in \Z: m \le x}$

As $x \in \sup \set {m \in \Z: m \le x}$, and there can be no greater $n \in \Z$ such that $n \in \sup \set {m \in \Z: m \le x}$, it follows that:

$x = \floor x$

$\blacksquare$