I am trying to prove the following:

If are mutually independent events, then

This is true for

by DeMorgan's Law

and since and are independent, and are independent

Therefore

Assume true for n

Then

Thus by the theorem for n=2

But by induction hypothesis

Thus

Which is what we want.

I was wondering if I proved the base case correctly and applied the induction hypothesis correctly. Thank you for your help.