Rational Numbers are Countably Infinite/Proof 3

Theorem
The set $\Q$ of rational numbers is countably infinite.

Proof
For each $n \in \N$, define $S_n$ to be the set:


 * $S_n := \left\{{\dfrac m n: m \in \Z}\right\}$

By Integers are Countably Infinite, each $S_n$ is countably infinite.

Because each rational number can be written down with a positive denominator, it follows that:


 * $\forall q \in \Q: \exists n \in \N: q \in S_n$

which is to say:
 * $\displaystyle \bigcup_{n \mathop \in \N} S_n = \Q$

By Countable Union of Countable Sets is Countable, it follows that $\Q$ is countable.

Since $\Q$ is manifestly infinite, it is countably infinite.