I suppose $2 - \sqrt{2} $ is rational. so $$2- \sqrt{2} = {a/b} $$ where a,b are integers and gcd(a,b) = 1.
$$\text{Step 1. } 2 = (a/b)^2 \text{ //squared both sides }$$ $$\text{Step 2. } 2b^2 = a^2 \text{ //We see $a^2$ is even }$$ $$\text{Step 3. }2b^2 = (2k)^2$$ Step 3 since $a^2$ is even and so $a$ is even too. Also, k is an integer. $$\text{Step 4. }b^2 = (2k)^2 \text{ //We see $b^2$ is even }$$ Step 4 since $b^2$ is even and so $b$ is even too.
We see $ a$ and $b$ are even. We see $gcd(a,b) \neq 1$
Is this proof correct ?