proof of summation of event probability

Question

Consider an experiment with sample space $\Omega$ and events $E_i \subseteq \Omega, \forall i\in \mathbb{N}$. Prove that if

$$\sum^n_{i=1}P(E_i)>1$$ then there must exist at least two events that have a non-zero probability occurring together.

How do I begin a proof like this?

Answer 1

Hint:

Inclusion-Exclusion Principle implies that

$Pr(E_1\cup E_2\cup E_3\cup E_4\cup\dots)=Pr(E_1)+Pr(E_2)+\dots-Pr(E_1\cap E_2)-Pr(E_1\cap E_3)-\dots +Pr(E_1\cap E_2\cap E_3)+\dots \pm\dots$

We further know that the probability of any event is at most $1$.

Try to prove the contrapositive statement: If $Pr(E_i\cap E_j)=0$ for every $i\neq j$ then $\sum\limits_{i=1}^nPr(E_i)\leq 1$