I want to prove that $\sqrt{2}$ is irrational. I suppose that $\sqrt{2}$ is rational. The square of an even number is an even number, same for the odds one. Then $\sqrt{2}$ can be written like this : $$\sqrt{2} = 2 \cdot k \quad k \in \mathbb{Z}$$ Also $\sqrt{2}$ is rational thus can be written like this : $\sqrt{2} = \frac{1}{b} \cdot a \quad a,b \in \mathbb{Z}^*$
$\frac{1}{b}$ must be equal 2, thus $b =\frac{1}{2}$. But $b$ is not rational, contradiction. $\sqrt{2}$ is irrational.