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.
