TITLE: How to prove Boole’s inequality
QUESTION [6 upvotes]: I am trying to prove Boole’s inequality
$$P\left(\ \bigcup_{i=1}^\infty A_i\right) \leq \sum_{i=1}^\infty P(A_i).$$
I can show it of any finite $n$ using induction. What to do for $\infty$ ?

REPLY [4 votes]: You can write out the infinite union as 
$$ \bigcup_{n=1}^\infty A_n = A_1 \cup (A_2 \cap A_1^c) \cup (A_3 \cap A_1^c \cap A_2^c)\cup \ldots $$
Each of these sets is disjoint, so you can use $\sigma$-additivity. 
Now just use the fact that the $i$th term is a subset of $A_i$, and so the probability of the $i$th term is less than or equal to the probability of $A_i$. 
Disjointness is your best friend in proving this sort of thing. 
There was a typo. It should be the complement of A_1, not A_2 in the second term.