Let $\mathbb{A}_{\mathbb{R}}^n$ be $\mathbb{R}^n$ endowed with the Zariski topology, where closed sets are algebraic sets (in $\mathbb{R}^n$) defined by real polynomials.

Suppose $V \subseteq \mathbb{A}_{\mathbb{R}}^n$ is an irreducible affine variety. Let $U$ be an open (with respect to the usual topology) ball $U$ around a non-singular point of $V$ and of small enough radius.

Does it then follow that the Zariski closure of $(V \cap U)$ is $V$? I thought it should be true (maybe not?), but I was wondering how I can show this. Any comments are appreciated. Thank you!