If infinity were odd then two times infinity would be even, but two times infinity is infinity, which is defined as odd. Therefore, infinity is not odd.
Another win for proof by contradiction - assume a premise, show how the premise leads to a contradiction, which concludes that the premise is false, therefore, the negation of your premise is true.
Let me show you:
prove that the number 12 is even.
Assume: 12 is not even
if 12 is not even, then there is no integer that doubles to 12. that is, 2x = 12 doesn't exist for an integer x.
However, when 6 is doubled the answer is 12. 2 x 6 = 12
therefore, our assumption is false and 12 is even
This process is also called reductio ad absurdum in logic. We are reducing the absurd by assuming the opposite of what we are trying to prove. This type of proof works well when we are working with things that go on forever. Since I can never get to the end of the integers, or prime numbers, I can prove that they are infinite by assuming that there is an end.
This prove process shows us not only that infinity is not odd, but also that it is not even. Which begs the question: Is infinity a number at all?
Actually, no it isn't. It is a concept which points us to the God of the universe!