I've been trying to prove the following statement:
Let $ f:U \rightarrow \mathbb{C} $ be holomorphic with $ \overline{B(0, R)} \subset U$. Suppose $ r < R $. Prove that $$ \sup\limits_{r \leq |z| \leq R}\left(\left|f(z) - \frac{1}{z}\right| \right) \geq \frac{1}{R}$$
It's a way of proving that $ \frac{1}{z} $ is not a uniform limit of a sequence of polynomials on a ,,ring'' $ r \leq |z| \leq R $.
I tried to use Cauchy's integral theorem to evaluate $ f(z) $, however it didn't lead me anywhere close to the solution.
I would appreciate some help with this exercise