Limit of density points is a density point?

Question

Let $\lambda(A)>0$ for some $A \subset \mathbb{R}^n$ and let $x_n \in A$ be a sequence of Lebesgue density points of $A$ with $x_n \to x \in int(closure(A))$. Must $x$ be a density point as well?

Answer 1

Hint: consider $A=[0,1]\cup {\bf Q}\subseteq {\bf R}$.