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.

My good sir! We need to show:
$$\left(\bigcup_{n\ge 1}A_n\right)'\subseteq\bigcap_{n\ge 1}A_n'$$ and 
$$\bigcap_{n\ge 1}A_n'\subseteq\left(\bigcup_{n\ge 1}A_n\right)'.$$
The following are all biconditional statements. See, if $x∈\left(\bigcup_{n\ge 1}A_n\right)'$, that is, if x is in the complement of all these sets collectively, then x certainly isn't in the union of all the sets, so we see $$x∉\left(\bigcup_{n\ge 1}A_n\right).$$ Now, if x is not in the union of all these indexed sets, then most certainly it is also not in the intersect of all these sets, or: $$x∉\left(\bigcap_{n\ge 1}A_n\right).$$ Then for a final step, another logical statement: if x is not in the intersection of all these indexed sets, then it is in the intersection of everything else (draw 3-part Venn-Diagrams to convince yourself this) thus, $$x∈\left(\bigcap_{n\ge 1}A_n'\right).$$
