In my textbook, the Bolzano-Weierstrass Theorem is stated as:
Bolzano-Weierstrass Theorem: A bounded sequence $\left(x_{n}\right)\in\mathbb{R^{n}}$ has a convergent subsequence.
Does this hold true for metric spaces $\left(\mathbb{R^{n}},d\right)$, where $d$ is an arbitrary metric? Or does $d$ have to be the standard Euclidean metric?