Infinite Product of Analytic Functions is Analytic

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

Also see

 * Derivative of Infinite Product of Analytic Functions