Which method can be used to prove "The sum of two even integers is always even"?

A direct proof is most appropriate here. Let two even integers be 2m and 2n. Their sum is 2m + 2n = 2(m+n), which is clearly divisible by 2 and hence even. This straightforward demonstration starting from definitions and reaching the conclusion is the hallmark of a direct proof. Induction is for statements indexed by natural numbers, contradiction assumes the opposite, and contrapositive proves the contrapositive form.