Prove the Bonferroni inequality

Hi,

I would like to know how to prove the Bonferroni inequatlity (I have actually started but at some point, I don't know how to prove).

http://s3.amazonaws.com/answer-board...3750009902.gif 1- P(A1) - P(A2)- P(A3)

Note: http://s3.amazonaws.com/answer-board...2968750373.gif

Proof:

http://s3.amazonaws.com/answer-board...9625000075.gif

= 1 - P(A1 U A2 U A3)

= 1 - P(A1) - P(A2) - P(A3) + P(A1 n A2)

+ P(A1 n A3) + P(A2 n A3)

- P(A1 n A2 n A3) (1)

http://s3.amazonaws.com/answer-board...6718750725.gif 1- P(A1) - P(A2)- P(A3) (2)

I would like to know how to justify, proof that

P(A1 n A2) + P(A1 n A3) + P(A2 n A3) - P(A1 n A2 n A3) is positive?

Like how do I go from the step before last step to my last step, that is from (1) to (2)?

Thank you