Problem: Let $\{ f_n \} $ be a sequence of analytic functions defined of a domain $U \subseteq \mathbb{C} $. Let $f_n \to f$ uniformly. Prove that $f$ is analytic on $U$.
Solution: We will apply Morera's theorem. Let us recall the Morera's theorem which is in some sense converse of Cauchy's theorem.
(Morera's Theorem) Let $U$ be an open connected set and $f: U \to \mathbb{C} $ be continuous. Suppose that
\[
\int _\gamma f(z) \mathrm{d} z = 0
\]
for all simple closed curve $\gamma $ in $U$. Then $f$ is analytic on $U$.
We need to prove that $f$ is analytic. Let $\gamma $ be any simple closed curve in $U$. Since $f_n$'s are analytic, and hence continuous. As the uniform limit of continuous functions are continuous, $f$ is continuous. The integral
\begin{align*}
\int_\gamma f_n(z) \mathrm{d} z = 0 & \implies \lim_{n \to \infty} \int_\gamma f_n(z) \mathrm{d} z = 0 \\
& \implies \int_\gamma \lim_{n \to \infty} f_n(z) \mathrm{d} z = 0 \\
& \implies \int _\gamma f(z) \mathrm{d} z = 0.
\end{align*}
Therefore, by Morera's theorem $f$ is analytic.