Let $I$ be a finite set with $|I|=n$ and $\forall i\in I $ let $a_i \in \mathbb{R}$. Prove by induction
$$\prod_{i\in I} (1+a_i)=\sum_{J\in \mathfrak{P}(I)}\left(\prod_{j\in J}a_j\right)$$
My doubts:
a) I think the RHS should be one more.
b) What does $J=\emptyset$ imply for the RHS.
Note:
I don't want the induction proof. Just clarification of these doubts.