Trying to solve exercise 1.1.18 in D.W. Stroock, Probability Theory, I somehow don't see how to get the hint in that exercise.
Given a set $\Omega$, a tail $\sigma$-algebra $\tau$ generated by $\sigma$-algebras $\cal F_n$, where each $\cal F_n$ is again generated by a set $A_n$, that is ${\cal F}_n = \{\emptyset, \Omega, A_n, \Omega \setminus A_n\}$. The sets $A_n$ are independent.
An atom $C\in \tau$ is a non-empty set which has no non-empty subset in $\tau$: If $\emptyset \neq B \subset C$ and $B\in \tau$, then $B=C$.
Now, if $C$ is an atom in $\tau$ then it can be written as a $\liminf$, more precisely: $C$ is an atom only if one can write $ C = \liminf_n C_n = \bigcup_n \bigcap_{k\geq n} C_k \quad \mbox{ where } C_n \in \{A_n, \Omega\setminus A_n\} $
Is that implication simple? I simply don't see it.
Thanks for any help.