How would you prove that the square root of two is irrational?

I think there are multiple ways of doing this, however I like proof by contradiction, I'll outline it as best I can below:

Let us suppose that sqrt(2) is rational. Therefore it can be written as x/y where x and y have no common factors (because if they did, the fraction could be cancelled).

Squaring both sides, this means 
Rearranging, we can say 

Then there are a few logical steps.
Since y is multiplied by 2, x^2 must be even. So the only way this can be the case is if x is even (since squaring an odd number still gives an odd number).
But since it is squared, that also means it is therefore divisible by 4, and so this means that y^2 and hence y must also be even.
This contradicts our assumption that they have no common factors, which means it cannot be rational.

I'll say voice is the strongest because it give you your personal opinion on things,and your voicing your reasoning..hope that ,if not let me know what i can do :