Prove that for all integers m and n, if m and m - n are odd, then n is even.
I'm not really sure where to start.
I would start by considering that an odd number is always equal to an even number plus 1. So adding two odds is the same as adding two evens plus (1+1). We're almost done - all you need do now is show that adding two evens plus 2 always yields an even - can you take it from here?
There is no requirement that n = m-1, so if m = 2k1+1, you don't know whether n = 2k1 or not.
I'm afraid I misread the original question - I thought it was asking to prove that the sum of two odds is even. But we can still use some of what I wrote in my first post. You know that m-n = odd, so that means n = m - odd, and you are told that m is odd. Now prove that subtracting two odds yields an even, in a similar manner as I suggested previously.