Complex-analytic implies holomorphic

Statement

Suppose $U\subset \mathbb{C}$ is a domain and $f:U\to \mathbb{C}$ is a complex-analytic function: for every point $z_0\in U$, there exists a real number $r>0$ and a power series $\sum _{n=0}^{\mathrm{\infty }}{a}_n(z-z_0{)}^n$ 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

We guess that the power series actually represents the Taylor expansion