Power Series Converges Uniformly within Radius of Convergence

Theorem
Let $\displaystyle S := \sum_{n \mathop = 0}^\infty a_n \left({x - \xi}\right)^n$ be a power series about a point $\xi$.

Let $R$ be the radius of convergence of $S$.

Let $\rho \in \R$ such that $0 \le \rho < R$.

Then $S$ is uniformly convergent on $\left\{{x: \left|{x - \xi}\right| \le \rho}\right\}$.