Given an arbitrary collection $A_n$of sets of the set $E$, does the relation lim sup$A_n=$lim inf $A_n$ implicates that the sets $A_n$ are necessarily increasing or decreasing in the sense of the union or intersection respectively ? Can you propose a proof ?
I know that the other way round the assertion is true, i.e. if $A_n$ is increasing or decreasing, then lim sup$A_n=$lim inf $A_n=:$lim $A_n.$
Thanks for your comment.
By definition $\liminf A_n\subseteq\limsup A_n$ so the statement $\liminf A_n=\limsup A_n$ is the same as $\limsup A_n\subseteq\liminf A_n$.
Actually it states that for every element $x$ for which the set $\{n\in\mathbb N\mid x\in A_n\}$ is infinite there exists an integer $n_x$ such that $n\geq n_x\implies x\in A_n$.
Observe that for every fixed integer $m$ the sets $A_1,\dots,A_m$ do not affect the answer on the question whether this statement is true or false. They "have nothing to say" about the set $\{n\in\mathbb N\mid x\in A_n\}$ being infinite or not, and they "have nothing to say" about the existence of such integer $n_x$.
This is enough already to conclude that the statement $\liminf A_n=\limsup A_n$ will not imply increasing or decreasing of the sets $A_n$.