Extremely difficult induction proof problem

I dunno who thought up this one, but it is clearly for masochists.

Prove that for all and all positive integers n

This MUST be proven by induction on .

In addition, these hints were provided for the question. They must be justified.

As I said, whoever came up with this one is a sadist.

I believe I have correct justifications on the two hints (unless I wrote something wrong), but I can't seem to provide the proof for a case (for proving by induction). Still, if anyone can help explain this one in detail, I'd be very appreciative.