If the events $\{E_n\}$ satisfy a certain property show that $P(\cap_{i=1}^k E_i) > 0$

If the events $\{E_n\}$ satisfy a certain property show that
$P(\cap_{i=1}^k E_i) > 0$

Let $\{E_n\}$ be events such that $\sum_{i=1}^kP(E_i) > k - 1$ then we
want to show that $P(\cap_{i=1}^k E_i) > 0$.
My approach for this problem is by contradiction. Suppose $P(\cap_{i=1}^k
E_i) = 0$ and then try and arrive at a contradiction. But we are not told
if the events are independent. I am not sure how from knowing
$P(\cap_{i=1}^k E_i) = 0$ to go to making conclusions about
$\sum_{i=1}^kP(E_i)$.