The irrationality of the square root of 2 Is there a proof to the irrationality of the square root of 2 besides using the argument that a rational number is expressed to be p/q?

If $\sqrt 2=p/q$ where $gcd(p,q)=1$ then $2=p^2/q^2$ <=>$p^2=2q^2$ and thus $p^2$ is even and thus $p$ is even let's say $p=2m$. Then $4m^2=2q^2$<=> $q^2=2m^2$  and thus $q$ is also even. So $gcd(p,q)>1$ which is false
