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 \preceq x \land x \preceq n}\right\} & : m \preceq n \\ \varnothing & : n \prec m \end{cases}$