The proof of an upper bound of a union of multiple sets?

Question

$\mathbb{P}\left(\bigcup_{i=1}^n A_i\right) \leq \min _k\left\{\sum_{i=1}^n \mathbb{P}\left(A_i\right)-\sum_{i: i \neq k} \mathbb{P}\left(A_i \cap A_k\right)\right\}$

I think mathematical induction is provable, but when k=n, I don't know how to prove it.