I thought this proof would be a straightforward direct proof. So, the contradiction would be "The sum of two integers is always even." Sparing the rigorous details: an integer n added to another integer leads to , which contradicts the statement, since is the representation of an odd number.
My teacher, however, proved this with two cases. The first case: a direct proof, using my strategy above, for . The second case, basically a similar proof to the one in the first case but now using . This second case is what has confused me. Isn't this step a bit redundant? Is it necessary? Does it enhance the proof, or just add superfluous information to it?