1. ## Bonferroni's inequality

I need to prove that, for any events $\displaystyle A_{k}, k=1,\ldots n$,

$\displaystyle \mathbf{P}\left(\cup_{k=1}^n A_k\right) \leq \sum_{k=1}^n \mathbf{P} (A_k) - \sum_{i<j}\mathbf{P} (A_i \cap A_j) + \sum_{i<j<k}\mathbf{P}(A_i\cap A_j \cap A_k)$

It is recommended to proceed by induction, but I cannot see how. If someone could outline some procedure I could follow to prove the more general case, it would be much appreciated.

e: corrected typo

2. you have equality if you change that sign to plus in the third sum, but then you need the prob of all four pairings and then subtract all five pairings...
you can see that if you draw a Venn diagram