Existence of an infinite countable compact subset of $\mathbf{R}$

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Studying compactness, I wondered whether a countable (and infinite) subset of $\mathbf{R}$ could be compact and I looked for examples of such sets, but I did not find anything. Is there any theorem or counter-example related to this question ?

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$$\{\,\tfrac 1n\mid n\in\Bbb N\,\}\cup\{0\} $$

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What about $\{\frac1n:n\in\mathbb N\}\cup \{0\}$?