Let n be an element ofZ. Prove that is odd if and only if is even.

Since it's a biconditional, I'm certain there will be two parts: 1) Proving , and 2) proving

It's in the proof by contrapositive section, so I'll try to prove it that way.

1) If is odd, then is even.

Contrapositive: If is odd, then is even.

Assuming is odd, then

for some integer m.

Therefore, is even and the implication is true.

My professor says that my proof is wrong, but I don't see how.