continuity of power series

Choose $z \in D$ and select $\delta >0$ so that $C = \overline{B}(z,\delta) \subset D$. Let $\rho = |z-x_0|+\delta$, note that $\rho < r_0$ and that $C \subset \overline{B}(x_0,\rho) \subset D$.

Then let $M_n = |a_n|\rho^n$, and note that $|a_n(x-x_0)^n | \leq M_n$, $\forall x \in C$, and $\sum M_n < \infty$.

Hence we can use the Weierstrass M-test to conclude that the series $\sum a_n (x-x_0)^n$ converges uniformly on $C$. Since each of the functions $\sum_{n<N} a_n (x-x_0)^n$ is trivially continuous, it follows from the uniform limit theorem that the limit function $x \mapsto \sum a_n (x-x_0)^n$ is continuous on $C$. Since $z\in D$ was arbitrary, the desired result follows.