How did the author arrive at the conclusion for $z >R$? For background information, I've included the full derivation of the solution at the bottom. I'm just stuck on that one little part though. How does $\frac{(z-R)}{|z-R|}-\frac{(-z-R)}{|z+R|}=2$??
Full solution for reference:
$$ \frac{z-R}{|z-R|}-\frac{-z-R}{|z+R|} = \frac{z-R}{|z-R|}+\frac{z+R}{|z+R|}$$
When $z>R$, $z-R$ and $z+R$ are both positive, giving two.
Similarly, it's zero when $z<R$.
The author never concluded $z>R$. He's given the answer for both cases: $z>R$ and $z<R$.