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:
 * $\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.

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

Such sources prefer to use the more conventional $\set {m, m + 1, \ldots, n}$, 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 $\closedint m n$ is to be interpreted as $\set {x \in \Z: m \le x \le n}$ or $\set {x \in \R: m \le x \le n}$, but it is recommended that the convention be specifically defined when it is used.