# 4.4: The Well-Ordering Principle

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The penultimate theorem of this chapter is known as the Well-Ordering Principle. As you shall see, this seemingly obvious theorem requires a bit of work to prove. It is worth noting that in some axiomatic systems, the Well-Ordering Principle is sometimes taken as an axiom. However, in our case, the result follows from complete induction. Before stating the Well-Ordering Principle, we need an additional definition.

Definition 4.35. Let $$A\subseteq \mathbb{R}$$ and $$m\in A$$. Then $$m$$ is called a maximum (or greatest element) of $$A$$ if for all $$a\in A$$, we have $$a\leq m$$. Similarly, $$m$$ is called minimum (or least element) of $$A$$ if for all $$a\in A$$, we have $$m\leq a$$.

Not surprisingly, maximums and minimums are unique when they exist. It might be helpful to review Skeleton Proof 2.90 prior to attacking the next result.

Theorem 4.36. If $$A\subseteq \mathbb{R}$$ such that the maximum (respectively, minimum) of $$A$$ exists, then the maximum (respectively, minimum) of $$A$$ is unique.

If the maximum of a set $$A$$ exists, then it is denoted by $$\max(A)$$. Similarly, if the minimum of a set $$A$$ exists, then it is denoted by $$min(A)$$.

Problem 4.37. Find the maximum and the minimum for each of the following sets when they exist.

1. $$\{5,11,17,42,103\}$$
2. $$\mathbb{N}$$
3. $$\mathbb{Z}$$
4. $$(0,1]$$
5. $$(0,1]\cap \mathbb{Q}$$
6. $$(0,\infty)$$
7. $$\{42\}$$
8. $$\{\frac{1}{n}\mid n\in\mathbb{N}\}$$
9. $$\{\frac{1}{n}\mid n\in\mathbb{N}\}\cup\{0\}$$
10. $$\emptyset$$

To prove the Well-Ordering Principle, consider a proof by contradiction. Suppose $$S$$ is a nonempty subset of $$\mathbb{N}$$ that does not have a least element. Define the proposition $$P(n):=$$$$n$$ is not an element of $$S$$" and then use complete induction to prove the result.

Theorem 4.38. Every nonempty subset of the natural numbers has a least element.

It turns out that the Well-Ordering Principle (Theorem 4.38) and the Axiom of Induction (Axiom 4.1) are equivalent. In other words, one can prove the Well-Ordering Principle from the Axiom of Induction, as we have done, but one can also prove the Axiom of Induction if the Well-Ordering Principle is assumed.

The final two theorems of this section can be thought of as generalized versions of the Well-Ordering Principle.

Theorem 4.39. If $$A$$ is a nonempty subset of the integers and there exists $$\ell\in \mathbb{Z}$$ such that $$\ell\leq a$$ for all $$a\in A$$, then $$A$$ contains a least element.

Theorem 4.40. If $$A$$ is a nonempty subset of the integers and there exists $$u\in \mathbb{Z}$$ such that $$a\leq u$$ for all $$a\in A$$, then $$A$$ contains a greatest element.

The element $$\ell$$ in Theorem 4.39 is referred to as a lower bound for $$A$$ while the element $$u$$ in Theorem 4.40 is called an upper bound for $$A$$. We will study lower and upper bounds in more detail in Section 5.1.

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