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Find a sequence of one-point-sets $A_n = \{\ell_n\}$ with $\ell_n\in\mathbb N$ for all $n\in\mathbb N$, such that $$\limsup_{n\to\infty} A_n=\mathbb N$$

I know the definition of the $\limsup$ of a sequence of sets, $$\limsup_{n\to\infty} A_n = \bigcap_{n=1}^\infty \bigcup_{k=n}^\infty A_k$$ I know the sequence has to contain each number of $\mathbb N$ infinitely often, but I'm not able to find a suiting sequence.

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  • $\begingroup$ 0,0,1,0,1,2,0,1,2,3,0,1,2,3,4,0,1,2,3,4,5,... $\endgroup$ – hmakholm left over Monica Oct 14 '13 at 20:55
  • $\begingroup$ $\ell_n = n\bmod \lceil\sqrt n\rceil$ $\endgroup$ – Hagen von Eitzen Oct 14 '13 at 21:13
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$1,1,2,1,2,3,1,2,3,4,1,2,3,4,5,\ldots$

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  • $\begingroup$ wow, that's really easy. I guess I thought way too complicated... thank you! $\endgroup$ – dinosaur Oct 14 '13 at 21:38

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