Real power series/Convergence/Continuous function/Fact

Let

${}f(x):=\sum _{n=0}^{\infty }c_{n}x^{n}\,$

be a power series and suppose that there exists some ${}x_{0}\neq 0$ such that ${}\sum _{n=0}^{\infty }c_{n}x_{0}^{n}$ converges.

Then there exists a positive ${}R$

(where ${}R=\infty$ is allowed) such that for all ${}x\in \mathbb {R}$ fulfilling ${}\vert {x}\vert <R$ the series converges absolutely. On such an (open) interval of convergence, the power series ${}f(x)$ represents a

continuous function.

Write a proof