Uniform limits of holomorphic functions are holomorphic

Statement

Let $U \subseteq \mathbb{C}$ be an open set and let $f_1, f_2, \dots$ be functions $U \to \mathbb{C}$ that are holomorphic on $U$. Let $f : U \to \mathbb{C}$ be a function such that $f_n \to f$ uniformly on each compact subset of $U$. Then $f$ is holomorphic on $U$. Moreover, $f_n' \to f'$ uniformly on each compact subset of $U$.

Remarks

Proof idea: $f$ is continuous. On each disc in $U$, Cauchy's integral theorem gives $\oint_{\partial T} f_n = 0$ for each triangle $T$, so $\oint_{\partial T} f = 0$ by the uniform convergence, and Morera's theorem applies. A main use: if a series $\sum g_k$ of holomorphic functions converges uniformly on compact subsets of $U$, then its sum is holomorphic on $U$. There is no real analogue. On $[-1, 1]$, the polynomials converge uniformly to $|x|$, which is not differentiable at $0$.

Explanations

L1 human

If holomorphic functions get close to a function $f$ uniformly, then $f$ is holomorphic too, and their derivatives get close to the derivative of $f$. This lets us build new holomorphic functions as limits of series, for example the zeta function $\sum n^{-z}$.