How to prove countably infinite? How do I prove the following set is countably infinite? 
$\{\frac{1}{n}: n\in\mathbb{Z}\setminus\{0\}\}$
I know that I can say this set is a subset of $\mathbb{Q}$, and that $\mathbb{Q}$ is infinite, thus this set is infinite. However, I've not yet proven that the rational numbers are countable, so I'm unsure how to proceed in proving this set countable. 
 A: Here is a one-to-one correspondence between the set you consider and the set $\{1,2,3,\ldots\}$ of positive integers:
$$
\begin{array}{cccccccccccccccccccccccccccc}
1, & -1, & 1/2, & -1/2, & 1/3, & -1/3, & 1/4, & -1/4, & \ldots \\
1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & \cdots
\end{array}
$$
A: A set X is countably infinite if there exists a bijection between X and $\mathbb{Z}$.
To prove a set is countably infinite, you only need to show that this definition is satisfied, i.e. you need to show there is a bijection between X and $\mathbb{Z}$.
Is there a bijection between $\left\{ \frac{1}{n} : n \in \mathbb{Z} \right\}$ and $\mathbb{Z}$?
A: Since there is a bijection $f\colon\mathbb N\to\mathbb Z\setminus\{0\}$ and the function $g\colon \mathbb Z\setminus\{0\}\to\{1/n:n\in\mathbb Z\setminus\{0\}\}$ defined by $g(n):=1/n$ is a bijection, we know that the composition $g\circ f$ is a bijection, so the set $\{1/n:n\in\mathbb Z\setminus\{0\}\}$ must be countable.
Remark. If you have no defined the set $\mathbb Q$, you can prove this statement using the set $\{1\}\times\mathbb Z\setminus\{0\}$. Then, when you have $\mathbb Q$, for instance, you can prove that this sets have the same cardinality with $\{m/n\in\mathbb Q:m=1\}$ which is a subset of $\mathbb Q$.
