The integers are countable

Define $f:\mathbb{N}\to\mathbb{Z}$ by
$$f(n)=\begin{cases}\dfrac n2&\text{if }n\text{ is even},\\[2mm]-\dfrac{n+1}{2}&\text{if }n\text{ is odd}.\end{cases}$$
The values start $0,-1,1,-2,2,-3,\dots$

On even numbers $n=2k$ we have $f(n)=k$, which takes every non-negative integer exactly once. On odd numbers $n=2k+1$ we have $f(n)=-(k+1)$, which takes every negative integer exactly once.

The two sets of values are disjoint and together cover $\mathbb{Z}$, so $f$ is a bijection.