Start for a proof of the fundamental theorem of algebra

Question

Let $p(z)=a_o+\dots+a_nz^n$, with $n\geqslant 1$ and $a_n\neq0$. Let $f:\mathbb{C}\rightarrow\mathbb{C}$ be defined by $f(z)=1/p(z)$.

I need to show that there is an $R>0$ such that \begin{equation}\left|z\right|>R\quad\rightarrow\quad\left|z^nf(z)\right|<2/\left|a_n\right|.\end{equation}

I have no clue how to do this...

Answer 1

Hint: We have $|\frac{a_kz^k}{a_nz^n}|\to 0$ as $|z|\to\infty$.