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Complete metric subspaces of $\mathbb{Q}$

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Is there a nice characterization for the complete metric subspaces of $\mathbb{Q}$ (with the usual metric)?

It seems like a such a subspace must have empty interior; if it contained an open interval then it would clearly contain a non-converging Cauchy sequence. But this isn't enough, because consider the subset $\left\{\frac{1}{n} : n \in \mathbb{N} \right\} \subseteq \mathbb{Q}$. This has empty interior but is not complete because it doesn't contain $0$. In other words, a complete subspace of $\mathbb{Q}$ must be closed (which I think is true about metric spaces in general).

So a complete subspace of $\mathbb{Q}$ must be closed and have empty interior, but are these two sufficient to imply completeness? If so, how do I prove this, and if not, what am I missing? I'm not necessarily looking for a complete answer, just some pointers in the right direction.


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