Totally bounded implies Cauchy subsequence

Totally bounded implies Cauchy subsequence: Let $(X,d)$ be a metric space that is totally bounded . Then every sequence $(x_n)$ in $X$ has a subsequence $(x_{n_k})$ that is a Cauchy sequence .

This is a key step in the compactness criterion “complete + totally bounded,” and it is closely related to constructing finite epsilon-nets at decreasing scales.