I am trying to better understand what it means for a sequence $A_n$ of subsets of a set $S$
to be such that $\bigcap^\infty_{n = 1} \bigcup^\infty_{m = n} A_n = \limsup A_n = \liminf A_n = \bigcup^\infty_{n = 1} \bigcap^\infty_{m = n} A_n$
I find the interpretation infinitely often and eventually always as in
\begin{equation} \bigcap^\infty_{n = 1} \bigcup^\infty_{m = n} A_n = \limsup A_n = \{ w \, \colon w \in A_n \quad \text{infinitely often} \} \end{equation} and \begin{equation}
\bigcup^\infty_{n = 1} \bigcap^\infty_{m = n} A_n = \liminf A_n = \{ w \, \colon w \in A_n \quad\text{eventually always} \} \end{equation}
very helpful and I am looking for an analogous interpretation what it means for the two to be equal.