Let $\{A_n\}$ a sequence of subsets of $\Omega$. Show:
$1_{\cup_n A_n} \le \sum_n 1_{A_n}$
If the sequence of sets are mutually disjoint, the inequality holds with equality. Now suppose not mutually disjoint. There exists $i, j \in \mathbb{N}$ such that $A_i \cap A_j \ne \emptyset$. Hence, the inequality would instead hold with strict equality.
How am I supposed to show that this inequality holds in general? Induction?