Rational Numbers are Countably Infinite/Proof 3

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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.

$\blacksquare$


Sources