Rational Numbers are Countably Infinite

Theorem
The set $$\mathbb{Q}$$ of rational numbers is countably infinite, that is, can be placed in one-to-one correspondence with the natural numbers $$\mathbb{N}$$.

Intuitive Proof
We arrange the rationals thus:

$$\frac 0 1, \frac 1 1, \frac {-1} 1, \frac 1 2, \frac {-1} 2, \frac 1 3, \frac 2 3, \frac {-1} 3, \frac {-2} 3, \frac 1 4, \frac 3 4, \frac {-1} 4, \frac {-3} 4, \ldots$$

It is clear that every rational number will appear somewhere in this list.

Thus it is possible to set up a bijection between each rational number and its position in the list, which is an element of $$\mathbb{N}$$.