Countability definition and S -> T -> Z -> N

Honestly I can't tell what the actual 'question' is here. This is almost by definition isn't it?

I'm sure that when they do this in class they must know that Z injects into N. So what really is the point of contention here? I wasn't sure but I figured it must be showing that S and T are equal right?

Sorry, I still don't like this.

The definition of countability is still wrong (injective vs bijective) at several place, in particular the first hint. I would only show injection of S->T->Z, and then remark that it is, in fact, a bijection. Don't show the Z->N bijection, that's just confusing. A short sentence stating this as an easy fact would be sufficient.

Not sure if this is better or not but I gave it a shot. Let me know what you think (Alternatively if you want to write it and send me an email - I'll get back to it asap)

Just had to change it lol - FINE... =)