Past exam of the mathematics course of the University of Cambridge / 2011 / ia / Paper 1 / 3F / a / Solution

The Bolzano-Weierstrass theorem states that every bounded sequence in $\mathbb{R}$ has a convergent subsequence. More generally, a bounded sequence in ${\mathbb{R}}^d$ has a subsequence converging to a point of ${\mathbb{R}}^d$.