Let $X$ be a set and $(A_n)_{n\geq 0},(B_n)_{n\geq 0}$ sequences of subsets of $X$, and suppose there exist both $A:=\lim_{n\to\infty} A_n$ and $B:=\lim_{n\to\infty} B_n$.
I'm asked to either prove or give a counterexample to one of the following:
- $A\cup B = \lim_{n\to\infty} (A_n\cup B_n)$
- $A\cap B = \lim_{n\to\infty} (A_n\cap B_n)$
- $A\Delta B = \lim_{n\to\infty} (A_n\Delta B_n)$
Now, the following statements are true:
$\liminf_{n\to\infty}(A_n\cap B_n)\subseteq \liminf_{n\to\infty}(A_n)\cap \liminf_{n\to\infty} (B_n)$
$\liminf_{n\to\infty}(A_n)\cup \liminf_{n\to\infty} (B_n)\subseteq \liminf_{n\to\infty}(A_n\cup B_n)$
where equality does not necessarily hold, and here is where I believe lies the possibility of finding a counterexample in regard to the first two affirmations.
It is not difficult to see that $B^c=\lim_{n\to\infty}(B_n^c)$ and $A^c=\lim_{n\to\infty}(A_n^c)$ (complement with respect to $X$), so it would seem as if the one equality thas is possible to prove is the one involving the symmetric difference, but I cannot prove the following statement: $$A\setminus B=\lim_{n\to\infty}(A_n\setminus B_n),\quad B\setminus A=\lim_{n\to\infty}(B_n\setminus A_n)$$
and even if I had it, since I´m not sure that the limit respects the union I have no clue how I could prove that in fact $A\Delta B = \lim_{n\to\infty} (A_n\Delta B_n)$ (if that were the case).
I'm currently stuck at this and I do not even know where to start looking for an easy counterexample.
Any help is appreciated :)