State Morera's theorem. Suppose $f_{n}(n=1,2, \ldots)$ are analytic functions on a domain $U \subset \mathbf{C}$ and that $f_{n}$ tends locally uniformly to $f$ on $U$. Show that $f$ is analytic on $U$. Explain briefly why the derivatives $f_{n}^{\prime}$ tend locally uniformly to $f^{\prime}$.

Suppose now that the $f_{n}$ are nowhere vanishing and $f$ is not identically zero. Let $a$ be any point of $U$; show that there exists a closed disc $\bar{\Delta} \subset U$ with centre $a$, on which the convergence of $f_{n}$ and $f_{n}^{\prime}$ are both uniform, and where $f$ is nowhere zero on $\bar{\Delta} \backslash\{a\}$. By considering

$$\frac{1}{2 \pi i} \int_{C} \frac{f_{n}^{\prime}(w)}{f_{n}(w)} d w$$

(where $C$ denotes the boundary of $\bar{\Delta}$ ), or otherwise, deduce that $f(a) \neq 0$.