Fundamental theorem of algebra from a minimum-modulus estimate

Let \(f:\mathbb C\to\mathbb C\) be entire and nowhere zero. For \(r>0\):

1. Show that, with the circle \(|z|=r\) traversed counterclockwise, \[ \int_{|z|=r}\frac{dz}{zf(z)}=\frac{2\pi i}{f(0)}. \]

2. Show that \[ \left|\int_{|z|=r}\frac{dz}{zf(z)}\right| \le \frac{2\pi}{\min_{|z|=r}|f(z)|}, \] and hence \[ \min_{|z|=r}|f(z)|\le |f(0)|. \]

Deduce the fundamental theorem of algebra by applying this to a monic polynomial and verifying \[ |a_0+a_1z+\cdots+a_{n-1}z^{n-1}+z^n| \ge |z|^n\left(1-\frac{|a_{n-1}|}{|z|}-\cdots-\frac{|a_0|}{|z|^n}\right). \]