Cantor intersection theorem

Cantor intersection theorem: Let $(X,d)$ be a complete metric space and let $(F_n)_{n\in\mathbb{N}}$ be a sequence of nonempty closed sets with

1. $F_{n+1}\subseteq F_n$ for all $n$, and
2. $\operatorname{diam}(F_n)\to 0$, where $\operatorname{diam}$ is the diameter in the metric $d$.

Then the intersection $\bigcap_{n\in\mathbb{N}} F_n$ consists of exactly one point.

This theorem is the metric-space analogue of the nested interval theorem and is a standard tool in fixed-point and completeness arguments.