# Real numbers/Nested intervals/Definition

Nested intervals

A sequence of closed intervals

in is called
(a sequence of)
* nested intervals*, if
holds for all
,
and if the sequence of the lengths of the intervals, i.e.

converges to .

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 ${}0$.