Nested closed bounded intervals intersect, and shrinking diameters force a unique point.Suggest a slogan
Let \([a_0, b_0]\supseteq[a_1, b_1]\supseteq\cdots\) be a decreasing sequence of closed bounded intervals in \(\RR\). Then \(\Intersect_n [a_n, b_n]\neq\emptyset\) [@Rud76], and if \(b_n - a_n\to 0\), the intersection is a single point.
Let \(X\) be a complete metric space and \(E_0\supseteq E_1\supseteq\cdots\) nonempty closed bounded subsets with \(\diam E_n\to 0\). Then \(\Intersect_n E_n\) consists of exactly one point [@Rud76].