Wiener's lemma

Suppose $\sum_{0}^{\infty }\left|c_{n}\right| < \infty , f(z) = \sum_{0}^{\infty } c_{n} z^{n},\left|f(z)\right| > 0$ for every $z \in \bar{U}$, and $1 / f(z) = \sum_{0}^{\infty } a_{n} z^{n}$. Prove that $\sum_{0}^{\infty }\left|a_{n}\right| < \infty$.