Definition:Closed Interval/Integer Interval

Definition
Let $S$ be the set $\N$ of natural numbers or $\Z$ of integers.

Let $\left({S, \le}\right)$ be the totally ordered set formed from $S$ and the usual ordering $\le$ on numbers.

Let $m, n \in S$.

The integer interval between $m$ and $n$ is denoted and defined as:
 * $\left[{m \,.\,.\, n}\right] = \begin{cases}

\left\{{x \in S: m \le x \le n}\right\} & : m \le n \\ \varnothing & : n < m \end{cases}$

Also denoted as
Some authorities consider $\left[{m \,.\,.\, n}\right]$ to be an abuse of notation, as there is nothing in it intrinsically to distinguish it from the closed real interval $\left\{{x \in \R: m \le x \le n}\right\}$.

Such sources prefer to use the more conventional $\left\{{m, m + 1, \ldots, n}\right\}$, but then again it relies upon the implicit understanding that the domain is the set of integers.

The context will frequently be sufficient to allow the reader to determine whether $\left[{m \,.\,.\, n}\right]$ is to be interpreted as $\left\{{x \in \Z: m \le x \le n}\right\}$ or $\left\{{x \in \R: m \le x \le n}\right\}$, but it is recommended that the convention be specifically defined when it is used.