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[1Z2](Solved on 2022-11-29) Given \(A_ 1,A_ 2\ldots \) sets , for \(nββ\), we define
\begin{eqnarray} \label{eq:limsup_ insiemi} \limsup _{nββ} A_ n {\stackrel{.}{=}}β_{n=1}^β β_{k=n}^β A_ k\\ {} \label{eq:liminf_ insiemi} \liminf _{nββ} A_ n {\stackrel{.}{=}}β_{n=1}^β β_{k=n}^β A_ k\\ {} \end{eqnarray}