Prove the sum of two odd numbers is even

Take two arbitrary integers, x and y

We can express the odd integer x as 2a + 1 for some integer a
We can express the odd integer y as 2b + 1 for some integer b

x + y = 2a + 1 + 2b + 1
x + y = 2a + 2b + 2

Factor out a 2:
x + y = 2(a + b + 1)

Since 2 times any integer even or odd is always even, then x + y by definition is even.