Cantor Intersection Theorem

Cantor Intersection Theorem: Let $(X,d)$ be a complete metric space and let $(F_n)$ be a sequence of nonempty closed sets such that $F_{n+1}\subseteq F_n \quad \text{for all } n,$ and $\operatorname{diam}(F_n)\to 0 \quad \text{as } n\to\infty.$ Then $\bigcap_{n=1}^\infty F_n$ consists of exactly one point.

This theorem generalizes the nested interval theorem from $\mathbb{R}$ to complete metric spaces and is a key completeness tool used in fixed point and approximation arguments.

Proof sketch (optional): Pick $x_n\in F_n$. Nestedness implies $(x_n)$ is Cauchy because $d(x_m,x_n)\le \operatorname{diam}(F_n)$ for $m\ge n$. Completeness gives $x_n\to x$. Closedness gives $x\in F_n$ for all $n$. Diameter $\to 0$ gives uniqueness.