I'm working my way through Martin Liebeck's "A Concise Introduction to Pure Mathematics", and one of the statements he asks the reader to prove is: $\forall x \in \mathbb{Z}$, $\exists m, n \in \mathbb{Z}$ such that $m \gt 0$, $n \gt 0$, and $x = m - n$.

The way I thought of going about this is to use a contrapositive proof since it feels easier to find counter-examples. So I rewrote the statement as: $\exists x \in \mathbb{Z}$, such that $\forall m, n \in \mathbb{Z}$, $m \gt 0$, $n \gt 0$, and $x \ne m - n$.

Now we proceed by cases:

When $m \gt n$ we subtract $n$ on both sides so we get $m - n \gt 0$. We take $x = 0$ so this case holds.

When $m = n$ we subtract $n$ on both sides to get $m - n = 0$. We take $x = 1$ so this case holds as well.

Finally, when $m \lt n$ we subtract $n$ to get $m - n \lt 0$. We take $x = 0$, and this holds as well.

We've proven that all three cases hold. Now because we've proven the contrapositive, the original statement holds as well.

My questions are:

* First, is this a valid proof?
* Second, is this actually a contrapositive?
* Third, this feels clunky. Is there a nicer way to prove the statement?

Thanks.