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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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Do you know that a subset of $\mathbb{R}$ is compact if and only if it's closed and bounded? Can you find an example of a countable set which is closed and bounded? –  Qiaochu Yuan May 12 '11 at 3:12
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Once one is aware of the example of a convergent sequence and its limit, an interesting follow-up is trying to understand what countable linear orders are possible as compact countable subsets of ${\mathbb R}$. –  Andres Caicedo May 12 '11 at 4:09
    
Thank you! So for instance an example would be: Let (sn)=1/n. Then it would be {1/n U 0}? –  hawaii99 May 12 '11 at 4:15
    
Yes, that would work. Is it clear to you why this is compact? –  t.b. May 12 '11 at 4:27
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Yes it is closed because it contains all the limit points and bounded since it is bounded below by 0 and above by 1. Then any closed and bounded subset of R^n is compact. –  hawaii99 May 12 '11 at 5:22

1 Answer 1

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

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