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Measurable sets

Let \{A_n\}_{n \in \mathbb{N}} be a sequence of measurable subsets of the real line which covers almost every every point infinitely often. Prove that there exists a set B\subset \mathbb{N} of zero density (\displaystyle \lim_{n\to \infty} \frac{|B\cap \{0,1,...,n-1\}|}{n}=0) such that \{A_n\}_{n \in B} also covers every point infinitely often.
Miklos Schweitzer 2009 Problem 8

I think I might have found a generalization of the problem. My approach is in the pdf attached.
Problem 8 – Miklos Schweitzer 2009

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