Proof by Mathematical Induction

In Chapter 3 we studied proof techniques that could be used in reasoning about any mathematical topic. In this chapter we'll discuss one more proof technique, called mathematical induction, that is designed for proving statements about what is perhaps the most fundamental of all mathematical structures, the natural numbers. Recall that the set of all natural numbers is ℕ = {0, 1, 2, 3, …}.

Suppose you want to prove that every natural number has some property P. In other words, you want to show that 0, 1, 2, … all have the property P. Of course, there are infinitely many numbers in this list, so you can't check one-by-one that they all have property P. The key idea behind mathematical induction is that to list all the natural numbers all you have to do is start with 0 and repeatedly add 1. Thus, you can show that every natural number has the property P by showing that 0 has property P, and that whenever you add 1 to a number that has property P, the resulting number also has property P. This would guarantee that, as you go through the list of all natural numbers, starting with 0 and repeatedly adding 1, every number you encounter must have property P. In other words, all natural numbers have property P. Here, then, is how the method of mathematical induction works.