[payporanymn]

You can order those integers but it's easier to sketch a proof if you let them be separate sets.

Say, Zp is the set of non-negative integers = {0, 1, 2...}, and Zn is the set of negative integers = {-1, -2, ...}. If you map the sets to the natural number set N, you get bijections. For Zp, you get a N -> Zp map, where element x from N maps to x-1 from Zp, as in:

1 -> 0
2 -> 1
3 -> 2
...

Then for Zn, you get a N->Zn map where x from N maps to -x from Zn, as in:

1 -> -1
2 -> -2
...

Since the set of integers Z = Zn + Zp, it's a union of two sets. Zn and Zp have been proven to be countable, so a union of them is also a countable set. I don't think including a proof of that would be needed?

Disclaimer: this is not official advice