Example of a sequence $a_n$ such that $\cap_{n \in \mathbb{N}} a_n = \emptyset$

Question

I'm looking for an example of a sequence $a_n$ consisting of non-empty, closed parts of $\mathbb{R}$ with $a_{n+1} \subset a_n$ for every $n$ such that $\cap_{n \in \mathbb{N}} a_n = \emptyset$, but I can not think of an example.

Also I need to find an example of a sequence $a_n$ consisting of non-empty, bounded parts of $\mathbb{R}$ with $a_{n+1} \subset a_n$ for every $n$ such that $\cap_{n \in \mathbb{N}} a_n = \emptyset$.

Answer 1

For the first part, $a_n=[n,\infty)$ works.

For the second part, a closed, bounded subset is compact, so by the finite intersection property of compact sets, there is no example.

As zipirovich points out in a comment, the second art of the problem actually doesn't say "closed".

Answer 2

For the second question, since there's no "closed" requirement, a sequence of open intervals such as $a_n=(0,1/n)$ works.