I am currently finishing up my Discrete Math course, and I just wanted to clear something up that has confused me for the past few days. My teacher posts answer keys to assigned homework problems online, and one of his recent solutions to one of these problems confused me. The problem was "Prove the following used the method of contradiction: The sum of two consecutive integers is always odd."
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 (n+1) leads to (2n+1), which contradicts the statement, since 2n+1 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 n + (n+1). The second case, basically a similar proof to the one in the first case but now using (n-1) + n. 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?