# An application of Weierstrass theorem for holomorphic functions

We have the following theorem for holomorphic functions.

Theorem (Weierstrass)

If $(f_n)$ is a sequence of holomorphic functions on an open set $U\subset\mathbb C$ such that $(f_n)$ tends uniformly to $f$ on every compact $K\Subset U$.

Then $f$ is also holomorphic and $({f_n}^{(k)})$ tends uniformly to $f^{(k)}$ for all $k$.

I would like to have a simple concrete application (without meromorphic functions, infinite products and elliptic functions) of this theorem if you know one. Thank you in advance.

A corollary of Weierstraß' theorem is that for every normal family $\mathscr{F}\subset \mathscr{O}(U)$ (where normality is $\mathbb{C}$-normality, i.e. we don't allow sequences converging locally uniformly to $\infty$), the family
$$\mathscr{F}^{(k)} = \{ f^{(k)} : f \in \mathscr{F}\}$$
of $k^{\text{th}}$ derivatives is again normal.