This sounds pretentious. Just say, "Let a be an integer".

You assumed that

and concluded that

is even. This means you proved the following: For every

,

implies

is even. You have not proved this: For every

,

is even. You cannot arbitrarily remove an assumption that was crucial in deriving the conclusion. This is similar to how the claim, "If you give me a million dollars, I can buy a golden Lamborghini" is not the same as "I can buy a golden Lamborghini".

Also, 4k - 2 above should be 4k + 2.

Edit: If 5 were congruent to 2 mod 4, then you would indeed reach a contradiction and the proof would be fine. That is, a = 5 would be a counterexample to the proved claim, "For every

,

implies

is even" because the premise of the implication is true (under our assumption, not in reality) and the conclusion is false. As it is, a = 5 does not contradict the proved claim.