Square Root Of 2 Is Irrational

Let $\sqrt{2} = \frac{p}{q}$ where p and q are two positive integers such that $\frac{p}{q}$ have no common factors.
$$ \left( \frac{p}{q}\right)^2=2 \\ \therefore p^2=2q^2 $$ This means that $p^2$ is an even number and so $p$ is an even number and can be written as $2r$. $$ \therefore 4r^2=2q^2 \\ q^2 = 2r^2 \\ $$ By the same argument as above this means $q$ can be written as $2s$ but then $\frac{p}{q}$ would have the common factor of 2, and so contradicting initial statement.

Therefore $\sqrt{2}$ must be an irrational number.