Probability – Inequality on Intersection of n Events

Question

How to prove following inequality on probability of intersection of $n$ events $A_i, i=1,2,3,\ldots,n$.
$$P\left(\bigcap_{i=1}^n A_i\right) \geq \left(\sum_{i=1}^n P(A_i)\right) -(n-1).$$
I have proved it by induction. But how it can be proved in some other way?

Answer 1

$$P\left(\bigcap_{i=1}^n A_i\right) = 1- P\left(\bigcup_{i=1}^n A_i^c\right) \\ \geq 1-\left(\sum_{i=1}^n P(A_i^c)\right) =1-\left(\sum_{i=1}^n (1-P(A_i))\right) = \left(\sum_{i=1}^n P(A_i)\right) -(n-1)$$

with equality when the $A_i^c$ are mutually disjoint, or at least any intersections have zero probability since then $P\left(\bigcup_i A_i^c\right) = \sum_i P(A_i^c) $