Sequence of partial sums converges locally uniformly?

Question

Suppose $f: U \to \mathbb{R}$ is a real analytic function defined by $f(x)=\sum_1^\infty a_n (x-x_0)^n$ and let $f_N=\sum_1^N a_n(x-x_0)^n$. Then $\{f_N\}$ converges to $f$ pointwise. Wikipedia says this convergence must be locally uniform, but I'm having trouble seeing why this is true.

Answer 1

If the series converges at some point $x$ with $|x - x_0| = r$, the terms must be bounded there, i.e. there is $B$ such that $|a_n| r^n \le B$ for all $n$. Then for any $p$ with $0 < p < r$ you get the series converging uniformly on $|x - x_0| \le p$, since $|a_n (x - x_0)^n| \le B (p/r)^n$.