Lower bound for complex polynomial beyond circle or radius R If we have a polynomial with $c_i$ a complex number
$$c_nz^n + c_{n-1}z^{n-1} + \cdots + c_1 z + c_0$$
then $$|P(z)| > \frac{|c_n|R^n}{2}$$
When |z| > R for some R
I have tried using the triangle inequality where I obtain,
$|P(Z)| \leq |c_n||z|^n + \cdots + |c_0|$. But I seem to keep getting stuck. Any hints on how to move forward? Thank you!

You have
$$\frac{|P(z)|}{|z|^n}
=\left|c_n+\frac{c_{n-1}}z+\frac{c_{n-2}}{z^2}+\cdots+\frac{c_0}{z^n}\right|.$$
Show that if $|z|$ is large enough, then
$$\left|\frac{c_{n-1}}z+\frac{c_{n-2}}{z^2}+\cdots+\frac{c_0}{z^n}\right|<\frac{|c_n|}2$$
etc.
