A typical setup for a Rouche's Theorem problem is one where you have a polynomial, say $f(z)$, defined on some domain $U$. For simplicity let's take $U$ to be the unit disk. What often happens is one splits up $f(z)$ into $g(z)$ and $h(z)$ such that $|g(z)| > |h(z)|$ for all $z \in U$, and then based on the zeros of $g(z)$ one can make conclusions about the number of zeros of $f(z) = g(z) + h(z)$.
What has been causing me some doubt is how the condition $|g(z)| > |h(z)|$ for all $z \in U$ is shown. Based on the examples I have seen one shows this inequality by using the triangle inequality and the maximum values of $g(z)$ and $h(z)$, which of course happens on the boundary of the domain. However, how do we know that the inequality still holds within the domain? Why does it suffice to only check the boundary?
An example: Find the number of roots of a polynomial using Rouche's Theorem.