If $|a_0| < 1$, then the complex polynomial $z^n + a_{n-1}z^{n-1} + ... + a_1 z + a_0$ has at least one root in the unit disc. I cannot use Rouche's Theorem here and I need a push in the right direction.

Write your polynomial as product of linear factors $\prod (z - \alpha_i)$. Then if all roots lie outside the unit disc you can see that $a_0 = \prod \alpha_i$ lies outside the unit disk. A contradiction.
