Prove DeMorgan's Theorem for indexed family of sets. Let $\{A_n\}_{n\in\mathbb N}$ be an indexed family of sets. Then:
$(i) (\bigcup\limits_{n=1}^\infty A_n)' = \bigcap\limits_{n=1}^\infty (A'_n)$
$(ii) (\bigcap\limits_{n=1}^\infty A_n)' = \bigcup\limits_{n=1}^\infty (A'_n)$
I went from doing simple, straightforward indexed set proofs to this, and I don't even know where to start.

$(i)$ let $ x \in (\cup_{n=1}^\infty A_n)' \Rightarrow x \notin A_n $ for some $n$. Therefore, $x \in A_n$ for some $n$. and hence, $x \in \cap_{n=1}^\infty (A'_n)$
Now, using same reasoning, prove the other direction: $$\cap_{n=1}^\infty (A'_n) \subseteq (\cup_{n=1}^\infty A_n)'$$ This would give you the desired result. Second part is analogous.
