Here is my rather simple idea. I will treat the set of real numbers as a set of discrete continuities, each separated by an Epsilon ball that tends to 0.
So, let's say P(b) is true. We then assume P(k) is true, and prove that P(k+e) is true, where e goes to zero.
I just want to know if this is a valid technique or not because our teacher said that mathematical induction can only be applied to discrete structures, but I see no difficulty in treating a continuous system as a set of infinitesimal discrete quantities. Mak
This seems unlikely, since there are plenty of bounded subsets of the real line having no maximum or minimum element. The natural numbers are well-ordered; this is induction. The reals are not.