I have an assignment in my text that asks me to "Show how induction can be used to conclude that $(A_1 \cup A_2 \cup \dots A_n )^c = A_1^c \cap A_2^c \cap \dots \cap A_n^c$. The issue I am facing is that I can prove DeMorgan's law for any $n$ without induction, and don't see why induction is necessary/possible here. Is it as simple as "let x belong to the complement of the union of $A_1$ through $A_n$ and assume the equation holds. Then if x belongs to the complement of the union $A_1$ through $A_{n+1}$, $x$ does not belong to $A_1, \dots, A_{n+1}$, thus belongs to each of their complements, thus is in the intersection of all of their complements." Is this the "induction?" Or would it be more correct to phrase it as "if x doesn't belong to $A_{n+1}$ then it belongs to its complement, and the equation holds?"
The other part of the question asks me to explain why induction can't be used to show that $\left (\bigcup_{n=1}^{\infty} A_n \right )^c=\bigcap_{n=1}^{\infty}A_n^c$. I am thinking it's because induction is valid only for a finite $n$, not infinity, but is there more to it? Thanks!!