Every bounded sequence in R^k has a convergent subsequence

Corollary (Bolzano–Weierstrass, sequence form): If $(x_n)$ is a bounded sequence in $\mathbb{R}^k$, then there exists a subsequence $(x_{n_j})$ and a point $x\in\mathbb{R}^k$ such that $x_{n_j}\to x.$

Connection to parent theorem: This is exactly the Bolzano–Weierstrass theorem , often recorded as a corollary once compactness /sequential compactness is being developed.