The integers are countable

Statement

The set $\mathbb{Z}$ of integers is countably infinite. The function $f \colon \mathbb{N} \to \mathbb{Z}$ with $f(2k) = k$ and $f(2k + 1) = -(k + 1)$ is a bijection.

Remarks

The bijection lists the integers as $0, -1, 1, -2, 2, \dots$. So $|\mathbb{Z}| = |\mathbb{N}| = \aleph_0$, although $\mathbb{N}$ is a proper subset of $\mathbb{Z}$.

Explanations

L1 human

The list $0, 1, 2, \dots$ never reaches the negative integers. Alternate between the two signs, and every integer gets a position in the list.