Mathematical Induction

Abstract

As with several topics in this book, our presentation of mathematical induction is somewhat different from standard approaches. We are not alone in realizing that understanding this very important method of proof requires, at the very least, that students work a variety of problems that go beyond the usual proof that a given finite series has a certain closed form. In addition to presenting problems arising from a wide range of situations, we provide examples in which various steps in the induction proof are not so obvious. In some, the base case is hard to determine; in others, it is not obvious how to set up the problem formally as a Boolean valued function of the positive integers corresponding to a proposition. There will be situations in which it is difficult to see what, in the problem, corresponds to the positive integer. Of course, we give a number of problems in which it is not so easy to prove the implication from n to n + 1.