uniform convergence of analytic function becomes analytic in a region

Question

Suppose that, $G$ is a region in the complex plane and $f_n : G \to \mathbb{C}$ is analytic for each $n\ge1$. Suppose further that, the sequence {$f_n$} converges uniformly on $G$ to a function $f:G \to \mathbb{C}$. Show that, $f$ is analytic on $G$

I need to the proof of the above result. Can somebody help me please.Thanks for your time.

Welcome to MSE! Do you have thoughts on the problem and can share the things you've tried? It helps to provide better guidance knowing those things. Regards — Amzoti, May 11 '13 at 06:32

score 1 · Answer 1 · answered May 11 '13 at 06:25

The key fact you need is Morea's theorem and fact that for $f_{n}\rightarrow f$ uniformly, $\int f_{n}=\int f$. The condition given is actually too strong (you just need $\int f_{n}\rightarrow \int f$). I hope this is enough for you to construct a proof by yourself.

uniform convergence of analytic function becomes analytic in a region

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