Page 274 - HOW TO PROVE IT: A Structured Approach, Second Edition

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Mathematical Induction

6.1. 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.

To prove a goal of the form ∀n ∈ NP(n):
First prove P(0), and then prove ∀n ∈ N(P(n) → P(n + 1)). The first of
these proofs is sometimes called the base case and the second the induction
step.

Form of the final proof:
Base case: [Proof of P(0) goes here.]
Induction step: [Proof of ∀n ∈ N(P(n) → P(n + 1)) goes here.]
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