Baire category related question

Question

Let $A_n$ be a sequence of closed sets of $\mathbb{R}$ such that $[a,b]\subseteq \cup A_n$, for some $a<b $. Show that at least one of the set $A_n$'s contains an interval.

Thanks in advance,

Answer 1

Hint: It has nothing to do with Urysohn's lemma. Use the Baire category theorem. Prove by contradiction. Suppose none of the $A_n$'s contains an interval, i.e., they are closed nowhere dense sets. Use Baire's theorem to show that the union of countably many closed nowhere dense subsets of $\mathbb R$ can't contain an interval. Exactly how was Baire's theorem stated in your class? If it was in terms of open dense sets, well, the complement of a closed nowhere dense set is an open dense set.