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Give an example of a compact countable infinite subset of $\mathbb{R}$. I'm having a difficult time, because I know that closed intervals $[a,b]$ are compact and infinite but are uncountable. Any help would be appreciated. Thanks!

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    @Alon: I know of partial results, but not a full answer. I think it is an interesting question.2011-05-12

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Try a convergent sequence together with its limit point.${}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}$