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 $|x|<r, r>0$ Assume that there exists a sequence $x_{n} rightarrow 0,left|x_{n}right|<r,$ 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 $|x|<r$?