Let $A:=[a_n,b_n],n\in\mathbb{N}$, be a nested sequence of closed intervals, i.e. $a_{n+1}>a_n$ and $b_{n+1}<b_n$ for all $n\in\mathbb{N}$. Show that the intersection $\cap_{n\in\mathbb{N}}A_n\neq \emptyset$ is non-empty. Moreover, if $\lim (b_n-a_n)=0$, then $\cap_{n\in\mathbb{N}}A_n=\{x_0\}$ consists of a single point. Is such a statement generally true for a nested sequence of non-closed intervals?
I have absolutely no idea how to do this one. Never worked with nested intervals before.