Complex-analytic implies holomorphic

Statement

Suppose $U \subset C$ is a domain and $f : U \to C$ is a complex-analytic function: for every point $z_{0} \in U$ , there exists a real number $r > 0$ and a power series $\sum_{n = 0}^{\infty} a_{n} (z - z_{0})^{n}$ such that the power series converges and agrees with $f$ in the ball of radius $r$ .

Then, $f$ is a holomorphic function: it is complex-differentiable, and the complex differential is a continuous function. In fact, $f$ is differentiable infinitely often.

Proof

To execute this proof, we need to argue that