So far, all of your observations are correct. Note to self: learn latex
To start, here are a few lemmas you will need, some of which you have already observed, others can be easily verified from the definition.
Lemma 1: if
By lemma 1, , and therefore
Base case: Let . Then the LHS is .
The RHS is . So LHS=RHS for our base case.
Inductive assumption: Suppose for some
Inductive leap: We want to show that ...