The random events \(A_{1}, A_{2}, \ldots, A_{n}\) are assumed to be independent. Show that

\[P\left(A_{1} \cup A_{2} \cup \cdots \cup A_{n}\right)=1-\left(1-P\left(A_{1}\right)\right)\left(1-P\left(A_{2}\right)\right) \cdots\left(1-P\left(A_{n}\right)\right)\]