I am trying to understand the justification for the inequalities in the proof of Lemma 3.1.3 described here.
The lemma is trying to prove that for any polynomial function $f : \mathbb{C} \to \mathbb{C}$ there exits a point $z_0 \in \mathbb{C}$ such that $|f|$ attains a minimum value in $\mathbb{R}$
The proof proceeds by letting
$f = a_0 + a_1 z + a_2 z^2 + \cdots + a_n z^n$ where $a^n \neq 0$ and $z \neq 0$
and
$A = max\{|a_0|, \cdots |a_n|\}$
Then
\begin{eqnarray} |f(z)| &=& |a_n| |z|^n \left|1 + \frac{a_{n-1}}{a_n} \frac{1}{z} + \cdots + \frac{a_0}{a_n} \frac{1}{z^n}\right|\\ &\geq& |a_n| |z|^n \left(1 - \frac{A}{|a_n|} \sum_{k=1}^{\infty} \frac{1}{|z|^k} \right) \\ \end{eqnarray}
The justification for this inequality is not clear to me. In particular the change from summation to subtraction is not obvious. If this is an application of the reverse triangle inequality, shouldn't the parenthesis be replaced by modulus?
Any help is appreciated.