An empty intersection of decreasing sequence of closed sets

What is an example of a family of closed subsets $F_0 \supset F_1 \supset F_2 \supset \dots $ of $\mathbb{R}$ so that $F_n \neq \emptyset$ for each $n$ and $\bigcap_{i=1}^n F_i = \emptyset$?

Thanks!

You should take $F_n=[n,+\infty)$ then the intersection is empty.

If $F_{n+1}\subseteq F_n$ then $\bigcap_{i=1}^n F_i = F_n$. This means that the family $\{F_n\mid n\in\mathbb N\}$ has the finite intersection property. In a compact space, this would mean that $\bigcap_{i=1}^\infty F_n\neq\varnothing$.

By that a decreasing sequence of sets whose intersection is empty it would have to be non-compact. In $\mathbb R$ this would mean that the sets are unbounded, so examples of the form $F_n=[a_n,+\infty)$ are essentially the only form of examples you can find (of course $(-\infty,b_n]$ is equally valid).