If a and b are two positive integers, prove $a^2-4b \neq 2$

I agree with Parcly Taxel's comment that it basically looks fine.

My only suggestion is to not use what you're trying to prove in each part as this makes it appear you've already shown, and thus don't need to go on, or you're assuming initially what you're trying to prove. Thus, for the first part of $a = 2k$, I would instead do something like $(2k)^2 = 4k^2 \equiv 0 \not\equiv 2 \pmod 4$.