\begin{definition}[Definition:Closed Interval/Integer Interval]
Let $S$ be the set $\N$ of natural numbers or $\Z$ of integers.
Let $\struct {S, \le}$ 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:
: $\closedint m n = \begin{cases}
\set {x \in S: m \le x \le n} & : m \le n \\
\O & : n < m
\end{cases}$
where $\O$ is the empty set.
\end{definition}