# sequence of sets with $\limsup A_n = \mathbb N$

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.

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

$1,1,2,1,2,3,1,2,3,4,1,2,3,4,5,\ldots$