# Real Number lies between Unique Pair of Consecutive Integers

## Theorem

Let $x$ be a real number.

### Right-open Interval

There exists a unique integer $n\in\Z$ such that:

$n \leq x < n + 1$

### Left-open Interval

There exists a unique integer $n\in\Z$ such that:

$n - 1 < x \leq n$

## Proof

### Existence

$S = \left\{{m \in \Z: m \le x}\right\}$

has a greatest element, say $n$.

Because $n+1>n$, $n+1\notin S$.

Thus $n+1> x$.

Thus $n\leq x < n+1$.

$\Box$

### Uniqueness

Let $n\in\Z$ be such that:

$n \leq x < n + 1$

We show that $n$ is a greatest element of the set:

$S = \left\{{m \in \Z: m \le x}\right\}$

so that the uniqueness follows from Greatest Element is Unique.

Because $n\leq x$, we have $n \in S$.

Let $m \in S$.

Because $m \leq x < n+1$, $n+1 > m$.

$n \geq m$.

Because $m$ was arbitrary, $n$ is a greatest element of $S$.

$\blacksquare$