# real analysis – Series of functions vanishing at a sequence in the region of convergence

Let $$f(x)=sum_{n=0}^{infty} a_{n} x^{n},$$ where the series converges for $$|x|0$$ Assume that there exists a sequence $$x_{n} rightarrow 0,left|x_{n}right| such that $$fleft(x_{n}right)=0$$ for all $$n$$. Prove that $$f(x)=0$$ for all $$x in(-r, r)$$

Here, can we use the uniform convergence of $$sum_{n=0}^{infty} a_{n} x^{n}$$ on $$|x|?