Is this true? I think I have a counter example. If we consider the set $(\mathbb{N},d)$, where $d$ is the discrete metric, then every subset closed ball is compact, but since $\mathbb{N}$ is infinite $(\mathbb{N},d)$ is not compact.
2026-04-03 09:10:51.1775207451
A metric space is compact iff every closed ball is compact
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