I would appreciate it if I could get some feedback on my attempt to proof De Morgan's Law. That is, if it is correct or if there is a better way to do this proof. I am self-studying probability theory from Grimmett and Stirzaker's book Probability and Random Processes and am going through the exercises. Thanks in advance.
Let ${A_i : i \in I}$ be a collection of sets.
$(\cup_{i} A_{i})^c = (A_1 \cup A_2 \cup A_3 \cup A_4 \ldots)^c$
$= (A_1 \cup A_2)^c \cup (A_3 \cup A_4)^c \cup \dots$
$= (A_{1}^c \cap A_{2}^c) \cup (A_{3}^c \cap A_{4}^c)\ldots$
$=(A_{1}^c \cap A_{2}^c \cap A_{3}^c \cap A_{4}^c \ldots)$
$=\cap_{i} A_{i}^c$
Best Answer
Perhaps simpler by definition:
$$x\in\left(\bigcup_i A_i\right)^c\iff x\notin\bigcup_i Ai\iff\;\forall\,i\;,\;x\notin A_i\iff$$
$$\iff \;\forall\,i\,,\;x\in A_i^c\iff x\in\bigcap_i A_i^c$$