Real numbers/Nested intervals/Definition

Nested intervals

A sequence of closed intervals

$I_{n}=[a_{n},b_{n}],\,n\in \mathbb {N} ,$

in ${}\mathbb {R}$ is called (a sequence of) nested intervals, if ${}I_{n+1}\subseteq I_{n}$ holds for all ${}n\in \mathbb {N}$, and if the sequence of the lengths of the intervals, i.e.

${\left(b_{n}-a_{n}\right)}_{n\in \mathbb {N} },$

converges to ${}0$.