$\bigcup_{k\in \mathbb{N}} F_k^\circ$ is dense in $\mathbb{R}$ > Let $\\{F_k\\}$ be a sequence of closed subset of $\mathbb{R}$ such that $\mathbb{R}=\bigcup_{k\in \mathbb{N}}F_k$. Then $\bigcup_{k\in \mathbb{N}} F_k^\circ$ is dense in $\mathbb{R}$. (Here $F_k^\circ$ denotes the interior of the set $F_k$). Using Baire's Category theorem, I can say that there exist at leat one $k\in \mathbb{N}$ such that $F_k^\circ $ is non-empty, but from here how will I conclude the result.

Assume for a contradiction that $\bigcup_{k\in \mathbb{N}} F_k^\circ $ is **not** dense in $\mathbb R.$ What that means is that there is an interval $I=[a,b]$ which is disjoint from $\bigcup_{k\in \mathbb{N}} F_k^\circ .$ It follows that $F_k\cap I$ is a nowhere dense closed set. So $I$ is the union of countably many nowhere dense closed sets. Can you take it from there?