Real Number is Integer iff equals Floor
Jump to navigation
Jump to search
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:
- $x = \floor x$
$\blacksquare$