Indexed set family/Natural numbers starting with n/Example

For ${}n\in \mathbb {N}$, let

${}M_{n}={\left\{x\in \mathbb {N} \mid x\geq n\right\}}\,$

be the set of all natural numbers that are at least as large as ${}n$. This is a family of subsets of ${}\mathbb {N}$ indexed by the natural numbers. We have the inclusions

$M_{n}\subseteq M_{m}{\text{ for }}n\geq m.$

The intersection

$\bigcap _{n\in \mathbb {N} }M_{n}$

is empty because there is no natural number that is above every other natural number.