(i) State the Baire Category Theorem for metric spaces in its closed sets version.

(ii) Let $f: \mathbb{C} \rightarrow \mathbb{C}$ be a complex analytic function which is not a polynomial. Prove that there exists a point $z_{0} \in \mathbb{C}$ such that each coefficient of the Taylor series of $f$ at $z_{0}$ is non-zero.