Nested interval theorem

Nested interval theorem: Let $(I_n)_{n\in\mathbb{N}}$ be a sequence of nonempty closed intervals in $\mathbb{R}$ such that $I_{n+1}\subseteq I_n$ for all $n$. Then $\bigcap_{n\in\mathbb{N}} I_n\neq\varnothing$.

More precisely, if $I_n=[a_n,b_n]$, then $\bigcap_{n\in\mathbb{N}} I_n=[\sup_n a_n,\inf_n b_n]$, using supremum and infimum . If additionally $b_n-a_n\to 0$, then the intersection is a single point.

This can be viewed as a special case of the Cantor intersection theorem in the usual metric on $\mathbb{R}$.