So I have to do an induction but I am not quite sure how to set it up. I have already proven that at each step I have either my intended result or I can advance one more step. I have also proven that there are a finite number of steps. Intuitively I have essentially completed the proof. I just can't figure out how to present this in a way that is completely rigorous. I get the feeling that "Well there can only be finitely many steps so eventually it will happen" is not good enough.