Uniform Limit of Analytic Functions is Analytic

Theorem
Let $U$ be an open subset of $\C$.

Let $\left\{ {f_n}\right\}_{n \mathop \in \N}$ be a sequence of analytic functions $U \to \C$ converging pointwise to $f: U \to \C$.

For each compact subset $K \subseteq U$, let $\left\{ {f_n}\right\}$ converge uniformly on $K$.

Then $f$ is analytic, and the sequence $\left\{ {f'_n}\right\}_{n \mathop \in \N}$ converges uniformly to $f'$ on all compact subsets of $U$.

Proof
By Equivalence of Local Uniform Convergence and Compact Convergence, $f_n$ converges to $f$ locally uniformly on $U$.

Then for any $z \in U$, there is an $\epsilon > 0$ so that:
 * $B_\epsilon \left({z}\right) \subset U$

and $f_n$ converges uniformly on $B_\epsilon \left({z}\right)$.

Let $\gamma$ be any simple closed curve in $B_\epsilon \left({z}\right)$.

Since $f_n \to f$ uniformly on $\gamma$ (because $\gamma \subset B_\epsilon \left({z}\right)$), we have:
 * $\displaystyle \lim_{n \to \infty} \int_\gamma f_n \left({z}\right) \, \mathrm d z = \int_\gamma f \left({z}\right) \, \mathrm d z$

Since each $f_n$ is analytic, we have that:
 * $\displaystyle \forall n \in \N: \int_\gamma f_n \left({z}\right) \, \mathrm d z = 0$

So we conclude also that
 * $\displaystyle \int_\gamma f \left({z}\right) \, \mathrm d z = 0$

Since $\gamma$ was arbitrary, we have by Morera's Theorem that $f$ is analytic in $B_\epsilon \left({z}\right)$.

Since $z$ was arbitrary, $f$ is analytic on all of $U$.

For the statement for the derivative, let $D$ be a disk of radius $r$ about $a$, contained in $U$.

We have Cauchy's Integral Formula for Derivatives


 * $\displaystyle f'_n \left({a}\right) = \frac 1 {2 \pi i} \int_{\partial D} \frac {f_n \left({z}\right)} {\left({z - a}\right)^2} \, \mathrm d z$

and:


 * $\displaystyle f'\left({a}\right) = \frac 1 {2 \pi i} \int_{\partial D} \frac {f \left({z}\right)}{\left({z - a}\right)^2} \, \mathrm d z$

Therefore:

Now the $f_n$ tend uniformly to $f$, and we can bound $r$ away from zero.

It follows that $f_n' \to f'$ uniformly in each compact disk contained in $U$.