generalized Bonferroni inequalities? Let $A_i \in\mathcal{A}$ be a sequence of events. Show that
$ 1. P(\cup_{i=1}^n A_i)\geq \sum_{i=1}^n P(A_i)-\sum_{i<j} P(A_i\cap A_j)$,
$2. P(\cup_{i=1}^n A_i)\leq \sum_{i=1}^n P(A_i)-\sum_{i<j}P(A_i\cap A_j)+\sum_{i<j<k} P(A_i\cap A_j\cap A_k).$
I have proved the first inequality, but the later seems much complicated to me?
 A: First prove it for $n=3$. Use the identity for 3 sets
$$
A_1\cup A_2\cup A_3=A_1\cup (A_2-(A_1\cap A_2))\cup(A_3-((A_1\cup A_2)\cap A_3))\tag1
$$
Note that 3 sets in the right are disjoint and $A_1\cap A_2\subset A_2$. So
$$
P(A_2-(A_1\cap A_2))=P(A_2)-P(A_1\cap A_2)\tag2
$$
Also
$$
A_3-((A_1\cup A_2)\cap A_3)=(A_3-(A_1\cap A_3))-((A_2\cap A_3)-(A_1\cap A_2\cap A_3))\tag3
$$
Note that 
$$
A_1\cap A_2\cap A_3\subset A_2\cap A_3$$
And 
$$
(A_2\cap A_3)-(A_1\cap A_2\cap A_3)\subset A_3-(A_1\cap A_3)
$$
 Thus by $(3)$
\begin{align}
P(A_3-((A_2\cap A_3)-(A_1\cap A_2\cap A_3)))&=P(A_3-(A_1\cap A_3))
\\&\quad\quad-(P(A_2\cap A_3)-P(A_1\cap A_2\cap A_3))
\\
&=P(A_3)-P(A_1\cap A_3)-P(A_2\cap A_3)+
\\&\quad\quad P(A_1\cap A_2\cap A_3)\tag4
\end{align}
Hence by $(1)$, $(2)$ and $(4)$
\begin{align}
P(A_1\cup A_2\cup A_3)&=P(A_1)+P(A_2)-P(A_1\cap A_2)+P(A_3-((A_1\cup A_2)\cap A_3))
\\
&=P(A_1)+P(A_2)-P(A_1\cap A_2) +P(A_3)
\\&\quad\quad -P(A_1\cap A_3)-P(A_2\cap A_3)+P(A_1\cap A_2\cap A_3)
\\
&=P(A_1)+P(A_2)+P(A_3)-P(A_1\cap A_2)
\\&\quad\quad -P(A_1\cap A_3)-P(A_2\cap A_3)+P(A_1\cap A_2\cap A_3)
\end{align}
Then use induction to prove the general case.
