Rational Numbers are Countably Infinite/Proof 1

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

Proof
The rational numbers are arranged thus:


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

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 $\N$.