Bolzano–Weierstrass theorem

Bolzano–Weierstrass theorem: In Euclidean space $\mathbb{R}^n$ with its standard metric, every bounded sequence $(x_k)$ has a subsequence that is convergent in $\mathbb{R}^n$.

This result is a core compactness phenomenon in Euclidean spaces and is closely tied to the Heine–Borel theorem and to sequential compactness .