Square Root of 2 is Irrational/Classic Proof

Proof
First we note that, from Parity of Integer equals Parity of its Square, if an integer is even, its square root, if an integer, is also even.

Thus it follows that:
 * $(1): \quad 2 \divides p^2 \implies 2 \divides p$

where $2 \divides p$ indicates that $2$ is a divisor of $p$.

Now, assume that $\sqrt 2$ is rational.

So:
 * $\sqrt 2 = \dfrac p q$

for some $p, q \in \Z$, and:
 * $\gcd \set {p, q} = 1$

where $\gcd$ denotes the greatest common divisor.

Squaring both sides yields:
 * $2 = \dfrac {p^2} {q^2} \iff p^2 = 2 q^2$

Therefore from $(1)$:
 * $2 \divides p^2 \implies 2 \divides p$

That is, $p$ is an even integer.

So $p = 2 k$ for some $k \in \Z$.

Thus:
 * $2 q^2 = p^2 = \paren {2 k}^2 = 4 k^2 \implies q^2 = 2 k^2$

so by the same reasoning:
 * $2 \divides q^2 \implies 2 \divides q$

This contradicts our assumption that $\gcd \set {p, q} = 1$, since $2 \divides p, q$.

Therefore, from Proof by Contradiction, $\sqrt 2$ cannot be rational.