Number of Roots Using Rouche Theorem

[ksuer]

Problem: find number of roots of $z^n + a_{n-1}z^{n-1} + ... + a_0, |z| < 1$

What is wrong with this argument:
Let f(z) = z^n + a_{n-1}z^{n-1} + \cdots + a_0, and g(z) = - a_{n-1}z^{n-1} - ... - a_0. Then, |f| > |g| and f+g = z^n. by Rouche thm, number of roots of f is equal to number of roots of f + g, which is n.

 

[Eynstone]

How do you get |f| > |g| ? Try the argument for f(z)=z^2 +2.