Proof of The Integer Lattice Admits an Enumeration by the Natural Numbers

Each result cited is universally quantified over the data in its own statement, and is applied here to the data named in the statement of this lemma.

Claim 1 ($\mathbb{Z}^{n}$ is countable). Let $T$ be the set of $n$-tuples in $\mathbb{Z}$, that is, the set of maps from $[n]$ to $\mathbb{Z}$. The set $\mathbb{Z}$ is countable by claim 1 of The Integers and the Rational Numbers are Countable, so $T$ is countable by claim 2 of Products and Powers of Countable Sets, applied to $\mathbb{Z}$ and to $n$.

Let $h:\mathbb{Z}^{n}\to T$ be the map sending $m$ to the map on $[n]$ whose value at $i$ is $m_{i}$; this is well defined because $m_{i}\in\mathbb{Z}$ for every $i\in[n]$ when $m\in\mathbb{Z}^{n}$, by Lattice-Periodic Functions and the Periodic Function Classes §lattice. It is injective: if $h(m)=h(m')$ then $m_{i}=m'_{i}$ for every $i\in[n]$, so $m=m'$ by claim 1 of Euclidean Points as Tuples of Real Numbers. By claim 5 of Basic Properties of Countable Sets, applied to the countable set $T$ and to $h$, the set $\mathbb{Z}^{n}$ is countable.

Claim 2 ($\mathbb{Z}^{n}$ is not finite). Let $\iota:\mathbb{N}\to\mathbb{R}$ be the canonical map of $\mathbb{R}$. By The Integers as a Subset of the Real Numbers one has $\iota(j)\in\mathbb{Z}$ for every $j\in\mathbb{N}$, and $0\in\mathbb{Z}$.

By claim 1 of Basic Properties of Initial Segments of the Natural Numbers one has $1\in[n]$. Let $f^{1}:\mathbb{N}\to\mathbb{R}$ be the map with $f^{1}(j)=\iota(j)$, and for $i\in[n]$ with $i\ne1$ let $f^{i}:\mathbb{N}\to\mathbb{R}$ be the map with $f^{i}(j)=0$. By claim 3 of Euclidean Points as Tuples of Real Numbers, applied to the set $\mathbb{N}$ and to these maps, there is a map $g:\mathbb{N}\to\mathbb{R}^{n}$ with $g(j)_{1}=\iota(j)$ for every $j\in\mathbb{N}$ and $g(j)_{i}=0$ for every $j\in\mathbb{N}$ and every $i\in[n]$ with $i\ne1$. Every component of $g(j)$ is an integer, so $g(j)\in\mathbb{Z}^{n}$ by Lattice-Periodic Functions and the Periodic Function Classes §lattice, and $g$ may be regarded as a map from $\mathbb{N}$ to $\mathbb{Z}^{n}$. It is injective: if $g(j)=g(j')$ then $\iota(j)=g(j)_{1}=g(j')_{1}=\iota(j')$ by claim 1 of Euclidean Points as Tuples of Real Numbers, whence $j=j'$ by claim 7 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field.

By The Natural Numbers Are Not Finite §injection, applied to the set $\mathbb{Z}^{n}$ and to $g$, the set $\mathbb{Z}^{n}$ is not finite. Together with claim 1 this proves clause 1.

Claim 3 (Clause 2). By clause 1 the set $\mathbb{Z}^{n}$ is countable and not finite, so Enumeration of an Infinite Countable Set §countable, applied to it, supplies a bijection from $\mathbb{N}$ onto $\mathbb{Z}^{n}$.