Verification of a proof that the difference of two odd integers is not odd

Question

Prove or disprove the difference of two odd integers is odd.

Here was my answer:

$m = 2s+1$

$n = 2t+1$

$m - n = (2s+1) - (2t+1)$

$= 2s - 2t$

$= 2(s-t)$

I then wrote the following:

Since $2(s-t)$ is even this is disproved since $(s-t)$ is an integer.

I didn't even get half a point, so is this answer completely wrong?

Answer 1

You could have said:

Let $m=2k_1+1,\ k_1\in\mathbb Z$ and $n=2k_2+1,\ k_2\in\mathbb Z$. We have $m-n=2k_1+1-(2k_2+1)=2k_1-2k_2=2(k_1-k_2).$ Let $k=k_1-k_2.$ Then we have $m-n=2k,\ k\in\mathbb Z$ which implies that $m-n$ is even. Therefore, the difference of two odd integers is even.