SCHWEITZER 2009 Problem 8

Let $\{A_n\}_{n \in \mathbb{N}}$ be a sequence of measurable subsets of the real line which covers almost every point infinitely often. Prove, that there exists a set $B \subset \mathbb{N}$ of zero density, such that $\{A_n\}_{n \in B}$ also covers almost every point infinitely often. (The set $B \subset \mathbb{N}$ is of zero density if $\lim_{n \to \infty} \frac {\#\{B \cap \{0, \dots, n - 1\}\}}{n} = 0$ .)