Rational Numbers are Countably Infinite/Proof 4

Theorem

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

Proof

Let $Q_\pm = \set {q \in \Q: \pm q > 0}$.

For every $q \in Q_+$, there exists at least one pair $\tuple {m, n} \in \N \times \N$ such that $q = \dfrac m n$.

Therefore, we can find an injection $i: Q_+ \to \N \times \N$.

By Cartesian Product of Natural Numbers with Itself is Countable, $\N \times \N$ is countable.

Hence $Q_+$ is countable, by Domain of Injection to Countable Set is Countable.

The map $-: q \mapsto -q$ provides a bijection from $Q_-$ to $Q_+$, hence $Q_-$ is also countable.

Hence $\Q$ is countable.

$\blacksquare$

Sources

• 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $2.5 \ \text{(v)}$