I need to prove that, for any events $\displaystyle A_{k}, k=1,\ldots n$,

$\displaystyle \mathbf{P}\left(\cup_{k=1}^n A_k\right) \leq \sum_{k=1}^n \mathbf{P} (A_k) - \sum_{i<j}\mathbf{P} (A_i \cap A_j) + \sum_{i<j<k}\mathbf{P}(A_i\cap A_j \cap A_k)$

It is recommended to proceed by induction, but I cannot see how. If someone could outline some procedure I could follow to prove the more general case, it would be much appreciated.

e: corrected typo