Proof by contradiction!

Prove that the following statement is true: the sum of two odd numbers is an even number

Let a and b odd numbers and suppose that a+b=2r is even for some integer r.

Since a and b are odd numbers, exist t, w integers such that a=2t+1 and b=2w+1.

Then

a+b = (2t+1) + (2w+1) = 2t+2w+2=2 (t+w+1). Then a+b is of the form 2*k where k=t+w+1, and this show that a+b is a even number. But this contradicts the hypothesis that a+b is odd.

Note: It is not necessary to prove by contradiction. With a direct prove is enough.