Show that for any $\displaystyle n$ events $\displaystyle A_1 ,A_2 ,...,A_n$ in a finite probability space, we have $\displaystyle P(A_1 \cup A_2 \cup \cdot \cdot \cdot \cup A_n) \leq P(A_1) + ...+P(A_n)$
Have you ever heard of Boole's inequality?
http://www.andrew.cmu.edu/course/21-228/lec7.pdf
Yes but how is this done using only the following axioms (first principles)
1) $\displaystyle P(\emptyset)=0$
2)$\displaystyle P(\Omega)=1$
3)$\displaystyle P(A_1 \cup \cdot \cdot \cdot \cup A_n)=P(A_1) + ...+ P(A_n)$ for $\displaystyle n$ disjoint sets $\displaystyle A_1 , ...,A_n \subseteq \Omega$
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.