Section 1.4 Order Properties of the Real Numbers
We are almost ready to start some actual analysis, where the functions and sets will be defined on the real numbers. (As mentioned earlier, we will tacitly assume that numbers in this course are real numbers, unless we state otherwise.) In order to do this, we need some facts about real numbers, including the following Order Axioms previously promised.
Given \(a,b,c\in\R\text{:}\)
Axiom 22. Order Axiom 2.
Axiom 23. Order Axiom 3.
Axiom 24. Order Axiom 4.
If \(a\lt b\) and \(c\gt 0\text{,}\) then \(ac \lt bc\text{;}\) if \(c\lt 0\text{,}\) then \(ac\gt bc\text{.}\)
Axiom 25. Order Axiom 5.
Example 26.
Definition 27.
Given these axioms, we say that the real numbers \(\R\) are linearly ordered because “less than” and “greater than” have meaning.
Aside
Axiom 25 is sometimes called “Archimedean.” We call numbers greater than zero positive and those greater than or equal to zero nonnegative; there is a similar definition for negative and nonpositive.
Adding the order axioms to the most basic facts about addition, subtraction, and multiplication already can allow us to prove slightly more advanced facts.
Example 28.
Under these axioms, the various familiar relationships between positive numbers, negative numbers, addition, and multiplication are true—such as “a positive times a negative is a negative” and “a positive plus a positive is positive”. Choose a couple of these familiar facts to prove.
Here is another useful fact related to the axioms. It may seem just as expected, but is quite different in style.
Problem 29.
Prove that for any positive \(a\in\R\text{,}\) there is an integer \(N\) such that \(0\lt \frac{1}{N} \lt a\text{.}\)
Finally, there is a special function which should be introduced now.
Definition 30.
Given \(a\in\R\text{,}\) we define the absolute value of \(a\), denoted \(|a|\text{,}\) to be \(a\) if \(a\ge 0\) and \(-a\) otherwise.
Example 31.
Example 32.
Example 33.
Show that \(|a|^2 = a^2\text{.}\)
The next two statements are pretty useful, and best proved (for now) by cases, though they have more clever proofs, too.
Problem 34.
Show that \(|ab| = |a| |b|\text{.}\)
Problem 35. The Triangle Inequality.
Show that \(|a + b| \le |a| + |b|\text{.}\)
We’ve seen the triangle inequality before. A related statement is sometimes called the reverse triangle inequality.
Problem 36.
Show that \(|a - b| \ge ||a| - |b||\text{.}\)
Problem 37.
Assume that there is a positive element of the preimage of \(\set{2}\) under the function \(f(x) = x^2\) from the reals to the reals; that is, assume \(\sqrt{2}\) exists. Show \(1 \lt \sqrt{2} \lt 2\text{.}\)
We do assume here that in a proof transition or other experience you have seen that \(\sqrt{2}\) is not a rational number, so now we have an irrational between two rationals, by a previous example.
The following problems are a good challenge to generalize this. You may need some other typical facts from an introductory proof course.
Problem 38.
Prove that between any two distinct real numbers there is a rational number.
Problem 39.
Prove that between any two distinct real numbers there is an irrational number.
We now turn to the main matters under consideration in this course.
