Prove that if x is even and y is odd, then x+y is odd

Let x and y be integers. Prove that if x is even and y is odd, then x+y is odd.

x is even, therefore x=2k for some integer k.
y is odd, therefore y=2j+1 for some integer j.

by substitution into x+y=odd we get

2k + 2j + 1 = odd

2k + 2j + 1 = 2(k+j) + 1, which is odd.