Intersection of a sequence of closed intervals

Question

Is it possible to have a sequence of bounded closed intervals $I_{1},I_{2},...$ which are not necessarily nested, such that $$\bigcap_{n=1}^{N}I_{n}\neq\emptyset$$ for every $N\in \mathbb{N}$ and $$\bigcap_{n=1}^{\infty}I_{n}=\emptyset$$

Answer 1

For completeness, the OP (originally) did not specify if some $I_n$ is compact. If some $I_n$ is compact, see Jose's answer.

Otherwise, $$ I_n=[n,\infty) $$ is a set of closed intervals satisfying the given condition since $$ \bigcap_{n=1}^NI_n=I_N\not=\emptyset, $$ but whose intersection is empty.

Answer 2

Assuming that you meant that each $I_n$ is closed and bounded, then the answer is negative, because then $\left(\bigcap_{n=1}^NI_n\right)_{N\in\mathbb N}$ is a decreasing sequence of non-empty closed and bounded intervals and therefore its intersection is non-empty. And$$\bigcap_{N=1}^\infty\bigcap_{n=1}^NI_n=\bigcap_{n=1}^\infty I_n.$$