Infinite Product of Analytic Functions is Analytic

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Theorem

Let $D \subset \C$ be an open set.

Let $\left\langle{f_n}\right\rangle$ be a sequence of analytic functions $f_n: D \to \C$.

Let $\displaystyle \prod_{n \mathop = 1}^\infty f_n$ converge locally uniformly to $f$.

Then $f$ is analytic.

Proof

Follows directly from:

Partial Products of Uniformly Convergent Product Converge Uniformly
Uniform Limit of Analytic Functions is Analytic

$\blacksquare$