Show that for any events in a finite probability space, we have
Have you ever heard of Boole's inequality?
http://www.andrew.cmu.edu/course/21-228/lec7.pdf
Use google, proofs this common are usually all over the internet.
http://www.math.ntu.edu.tw/~hchen/te...s/lecture2.pdf
If you don't like that one there are several others...
Yes, I've seen a proof using this argument, wikipedias in fact.