[Math] Proving Fundamental Theorem of Algebra using Maximum Principle

Question

I'm trying to prove FTA by using the maximum principle. Here's what I did,

Let $P$ be a polynomial of degree at least $1$ and assume that $P$ has no zeros. Define $$f(z):=\frac{1}{P(z)}.$$ Then $f$ is holomorphic on the disk $|z| \leq R$. Since $f$ is continuous, it attains its maximum value for some complex number, say $w$. By the Maximum Principle, $w$ lies on the boundary and
$f(z) \leq f(w)$ for all $|z| \leq R$

How do I get a contradiction from here?

Thank you

Answer 1

You can't say $f(z) \le f(w)$, because these are complex numbers; what you want is $|f(z)| \le |f(w)|$.

Now use the fact that $|P(z)| \to \infty$ as $|z| \to \infty$.