Prove the following through proof by contradiction For all i

Prove the following through proof by contradiction

For all integers m and n, if m+n is even then m and n are either both odd or both even.

Let the number m be even = 2p

Let the number n be odd = 2q + 1

Let the number m+n will be even if one is odd and one is even

m + n = 2p + 2q + 1 = 2(p+q) + 1 = 2k1 + 1, which is an odd, hence the sum is odd

This is a contradiction since m+n is an odd, and i assumed that the sum of (m+n) is even

Hence the (m+n) is even then m and n are either both odd or both even