Indexed set family/Nested intervals/Example

Let ${}x$ be a real number, and let ${}x_{n}$ denote the rational number that consists of the digits of ${}x$ in the decimal system up to the ${}n$th digit after the point. We define the intervals

${}M_{n}=\left[x_{n},x_{n}+{\left({\frac {1}{10}}\right)}^{n}\right]\subset \mathbb {R} \,.$

These are intervals of length ${}{\left({\frac {1}{10}}\right)}^{n}$, and we have

${}\{x\}=\bigcap _{n\in \mathbb {N} }M_{n}\,.$

The family ${}M_{n}$, ${}n\in \mathbb {N}$, is a family of nested intervals for ${}x$.