Proving the convergence of $\sum\limits_{n=1}^{\infty}\frac{1}{1+z^n}$ for $|z| > 1$ $\sum\limits_{n=1}^{\infty}\frac{1}{1+z^n}$, $|z|>1$.
There are two facts that my professor uses that I am confused about.
The first is: $|1+z^n| \geq ||z|^n-1|$, I believe this is true for any $|z|$.
The other is: $\frac{1}{|z|^n-1} \leq \frac{2}{|z|^n}$, I believe this is also true for any $|z|$.
Can anyone prove these statements for me?

The first statement is a version of the triangle inequality.
$$\begin{align}
|x|&=|x-y+y|\le|x-y|+|y|\\
&\implies |x|-|y|\le|x-y|.
\end{align}$$
Similarly $|y|-|x|\le|x-y|$. Hence, 
$$||x|-|y||\le|x-y|.$$
The other inequality is equivalent to
$$|z|^n\le 2|z|^n-2$$
or
$$|z|^n\ge 2.$$
