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Is it possible to find a counter example to argue this:

$\limsup\limits_{n\rightarrow \infty} (A_n \cap B_n)$ = $\limsup\limits_{n\rightarrow \infty} A_n \cap \limsup\limits_{n\rightarrow \infty} B_n $

where $A_n$ and $B_n$ are two sequences.

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$$A_{2n}=B_{2n+1}=A\ne\varnothing,\qquad A_{2n+1}=B_{2n}=\varnothing$$ Still, an inclusion is always valid...

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