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Let $f_{n}$ be a sequence of holomorphic functions on an open set $U \in \mathbb{C}$, which tends to f uniformly on $U$. Prove that f is holomorphic on U

I have proved a simple case which is if $U$ is a simply connected domain, then a continuous function $g$ on $U$ is holomorphic if and only if $\int_{\gamma}g=0$ for any closed curve $\gamma$ in $U$.

But I don't know how to apply to the general case $U$ is an open set.

Thanks.

finiteness
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1 Answers1

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For a function to be holomorphic is a local property, i.e., to show that a function on $U$ is holomorphic on all of $U$, it's enough to show that every point in $U$ has an open neighborhood on which $f$ is holomorphic. But every point in $U$ has an open neighborhood that is a small disk inside $U$. Since disks are simply connected, what you've already proved is enough to finish the job.

Andreas Blass
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