# Union of closed sets in $\mathbb{R}^n$ feature

Let $F$ be nonempty closed set in $\mathbb{R}^n$ and $F=\bigcup\limits_{m=1}^{\infty}F_m$, where $F_m$ are closed sets. Prove, that there exists a number $m_{_0}$ and a closed ball $\bar{U}\subset\mathbb{R}^n$ such that $F\cap\bar{U}=F_{m_{\ _0}}\cap\bar{U}$ and $F\cap U\ne\varnothing$.

There is a proof in the book the problem is from, by contradiction and using a decreasing sequence of nested balls. But there is a place in the proof, that I find dubious. So, I should like to see, how the community solves the problem.

Since $F$ is a closed subset of a complete metric space it is complete.
By the Baire Category Theorem, one of the $F_m$ is NOT nowhere dense, say $F_{m_0}$.
But $F_{m_0}$, is closed so $F_{m_0}$ is NOT nowhere dense means it has non-empty interior. Take $U$ to be some open ball in the interior of $F_{m_0}$.
• The problem is essentially the statement of the Baire Category Theorem (BCT3 on wikipedia) in $R^n$ so I am doubtful that a "simple" solution exists. The sequence of decreasing balls is a standard way to prove the BCT. Dec 23, 2017 at 22:37