uniform convergence of derivative of the series of analytic functions

Question

Suppose $D\subseteq \mathbb{C}$ is a domain(open and connected set).$f_n:D\to \mathbb{C}$ is a sequence of analytic functions and $f:D\to \mathbb{C}$ is continuous.

(i) If $\sum f_n$ converges uniformly to $f$ on all compact subsets of $D$ then $f$ is analytic.

(ii)$f'(z)=\sum f'_n(z)$

Part (i) is an application if Morera's theorem.I did that.

For part (ii), We can use cauchy integral formula to write $f',f'_n$ in terms of $f,f_n$ respectively.then by a closed disk We can prove that $\sum f'_n$ converges to $f'$ pointwise.But how we prove that the convergence is uniform?

Answer 1

Hint

Use Cauchy’s differentiation formula and a compact that contains the circle of Cauchy’s formula on which the convergence of the series is uniform for $f$.