The usual way to deal with an infinite number of cases is to group them into a finite number of cases and then deal with those. That's how induction works. So i

The usual way to deal with an infinite number of cases is to group them into a finite number of cases and then deal with those. That's how induction works. So if you can prove that (1) the machine is in a certain condition at some step and (2) whenever the machine is in that condition at some step it will get back into that same condition at a later step, then you can prove that the machine will never halt.