Hello,

I am confused with the following question.

Suppose $\displaystyle E_1, E_2,..., E_n$ are events, then

$\displaystyle \sum_{i=1}^{n} P(E_i) - \sum_{i<j}P(E_i E_j) \leq P(\cup_{i=1}^{n}E_i) \leq \sum_{i=1}^{n} P(E_i)$

The right hand side is Boole's inequality. Prove the left hand side.

I am trying to prove this using induction, but I am not sure if I am doing it correctly.

For $\displaystyle n=2$,

Let $\displaystyle E_1$ and $\displaystyle E_2$ be events,

then, $\displaystyle \sum_{i=1}^{2} P(E_i) - \sum_{i<j}P(E_i E_j) = P(E_1) + P(E_2) - P(E_1 E_2) = P(E_1 \cup E_2)$.

Thus, this holds true for $\displaystyle n=2$.

Am I going on the right track?

Any suggestion for the inductive step?

Thanks in advance,