The Integer Lattice Admits an Enumeration by the Natural Numbers

Let $\mathbb{N}$ be the set of natural numbers, let $\mathbb{Z}$ be the set of integers, let $n\in\mathbb{N}$ (so that $1\le n$, by claim 4 of Properties of the Order on the Natural Numbers), and let $[n]$ be the initial segment determined by $n$. Let $\mathbb{R}^{n}$ be Euclidean space, whose points are read as maps on $[n]$ with real values, the component of $m\in\mathbb{R}^{n}$ at $i\in[n]$ being written $m_{i}$, and let $\mathbb{Z}^{n}$ be the integer lattice, the set of those $m\in\mathbb{R}^{n}$ with $m_{i}\in\mathbb{Z}$ for every $i\in[n]$. The notions finite, countable and bijection are those of the indicated definitions.

1. (Countability)¶ The set $\mathbb{Z}^{n}$ is countable and is not finite.

2. (Enumeration)¶ There is a bijection from $\mathbb{N}$ onto $\mathbb{Z}^{n}$.