It looks ok in general. The only nit-pick I have is the order of demonstrating n ->n+1. You kind of assumed n+1 was true then justified it after the fact. I would just reverse the order so that it flows more naturally going from n, doing some algebra, and implying n+1.
But yes, once you show n->n+1 you are basically done. You only have to make some sentence wrapping up the proof.