Suppose $E\_1,E\_2,\ldots , E\_i$ are independent events. prove by induction that $$P\left(\bigcup\limits\_{i=1}^{n} E\_i\right) = 1-\prod\limits\_{i=1}^{n}(1-P(E\_i))$$

The first step is easy. For $n=1$ we have
$$P\left(\bigcup\limits\_{i=1}^{1} E\_i\right) = P(E\_1)= 1-(1-P(E\_i))=1-\prod\limits\_{i=1}^{1}(1-P(E\_i))$$
I don't really see how to continue from here though. Any help would be appreciated!