Comments on Chapter 1

Page 3

The symbol N (for natural numbers) that appears on page 3 is not defined until page 7. Notice that there is a Notation Index at the back of the book on pages 679–680.
You should be aware that some mathematicians include the number 0 in the set of natural numbers. Watch out for this variation if you read other sources.
When writing by hand, one generally renders boldface symbols with double strokes. Thus N appears something like the following: \(\bbold{N}\). Some mathematics books now print such symbols with a special font, called “blackboard boldface”, that imitates the handwritten version of boldface.

Page 4

Note for honors students: The statement in the middle of page 4 that “you cannot increase area by reassembling pieces” is correct, but there is more to the story. According to the surprising “Banach–Tarski paradox”, a three-dimensional ball can be divided into finitely many pieces that can be reassembled into a ball of double the size. Of course, these pieces are not physically realizable: they are so-called nonmeasurable sets. The paradox shows that there are subtleties in defining the notion of volume. Hence there is a point to the effort that we will be making later in the course to define the integral rigorously.

Page 7

The following statement at the end of the Warning may appear strange: “If \(x \lt 0\), then by the Second Multiplicative Property, \(x \lt 1\) implies \(x^2 \gt x\).” Under the hypothesis that \(x \lt 0\), shouldn’t the condition that \(x \lt 1\) be a conclusion rather than a second hypothesis? The answer is yes, but only after page 9. On page 7, we do not yet officially know that \(0 \lt 1\). This “obvious” property is not part of the Order Axioms on pages 6–7, so it needs to be proved. After Example 1.2 on page 9, you can use this property without further mention.

Also on page 7, the author explains the notation Z for the integers in a parenthetical remark that “Zahlen is German for number”. More precisely, Zahl is number (singular), while Zahlen is numbers (plural).

Typographical note: Consistency of notation demands that in the definition of Z, the initial centered ellipsis dots be lowered to the baseline and followed by a comma.

Page 10

At the bottom of the page, the final sentence in Case 3 uses not only commutativity but also associativity.

Page 13

The wording of the second paragraph of the proof is inexact. It is not a valid deduction to say that “\(x\lt y+\epsilon\) for all \(\epsilon \gt 0\) in either case”, for the division into cases happened after the number \(\epsilon\) was specified. A correct statement is that “\(x\lt y+\epsilon\) for the specified \(\epsilon\) in either case, and since \(\epsilon\) is an arbitrary positive number, \(x\lt y+\epsilon\) for all \(\epsilon\gt 0\).”

The third paragraph of the proof starts by assuming the validity of the first clause of the equivalence and ends by concluding the validity of the second clause, so formally all that has been proved is one direction of the “if and only if” statement. The converse direction follows either by observing that all the steps are reversible or by rewording the argument to make each step an equivalence.

Similarly, the fourth paragraph of the proof covers only one direction of the “if and only if” statement. The converse direction, however, is already covered by Theorem 1.7i.

In the third paragraph from the bottom of the page, notice that the endpoints of the empty set are not well defined. It might be preferable to declare intervals to be nonempty as part of the definition of the word “interval”; otherwise one needs to say “nonempty open interval” in statements like Definition 3.12 on page 77.

Page 14

At line 3, for “belongs to the open intervals \( (-\epsilon,\epsilon)\) for all \(\epsilon \gt 0\)” read “belongs to the open interval \( (-\epsilon,\epsilon)\) for all \(\epsilon \gt 0\)”.

Page 15, Exercise 1.2.9

Everything preceding part b should be labeled as part a (the label a is missing).

After making the preceding emendation, observe that although part a says to “use Postulate 1 and Remark 1.1”, only Postulate 1 is needed. Remark 1.1 does not come into play until part b.

Page 16, Definition 1.10

Notice that the terminology “has a finite supremum” in part ii later gets abbreviated to “has a supremum” (for instance in Theorems 1.20 and 1.21 and in Exercise 1.3.8).

Remark 1.13 on page 17 should really be part of Definition 1.10ii: that remark is what makes the notation \(\sup E\) well defined. Moreover, the uniqueness of the supremum is used in Example 1.11 and in the paragraph preceding the example.
Notice that the singular form of the word is “supremum”, while the plural form is “suprema”.

Page 17

In the statement of Theorem 1.14, “\(\epsilon\gt 0\) is any positive number” is redundant; simply “\(\epsilon\) is any positive number” suffices.

Page 18

In Theorem 1.16, the boldface symbol N should be in an upright font (not slanted). The same comment applies to the statement of Theorem 1.22 on page 23.

Page 19

The statement in the first paragraph of the Strategy that “[i]f \(n=1\), then \(k_0 =0\)” is incorrect. When \(n=1\), the set \(E\) is empty, so \(k_0\), the supremum of \(E\), is undefined. The last paragraph of the Strategy makes explicit that \(n\) has to be sufficiently large for the set \(E\) to be nonempty.

Page 20

In line 2, it would be preferable to say “the supremum” rather than “a supremum”, since we know from Remark 1.13 on page 17 that the supremum is unique. Alternatively, in the context of this argument, it would suffice to say “an upper bound”.

Page 20

In accordance with the remark on page xii of the Preface, the “if” in part iii of Definition 1.19 means “if and only if”.
Notice that “infimum” is the singular form of the word, and “infima” is the plural form.

Page 21

In the last sentence on the page, an additional convention (implicit, but unstated) is that \(-\infty\lt\infty\).

Page 22, Exercise 1.3.6

In part b, for “is a nonempty” read “is nonempty”.

Page 23

Theorem 1.23, called the Principle of Mathematical Induction in Section 1.4, is called the Axiom of Induction on pages 623–624 in Appendix A.

In the fourth line of the proof of Theorem 1.23, the reference to Remark 1.1i should be to Remark 1.1ii.

Page 25

The binomial coefficient \( \binom{n}{k}\) is commonly pronounced “\(n\) choose \(k\)”, for it represents the number of combinations of \(n\) objects taken \(k\) at a time.

Page 27, proof of Remark 1.28

Notice that although the number \(m_0\) appears by applying Remark 1.27, its value is different from the \(m_0\) in the proof of Remark 1.27.

Page 28

Exercise 1.4.5 involves the \(n\)th root of a positive real number, a notion that has not officially been defined. One needs the completeness axiom to prove that a positive real number does have a (unique) positive \(n\)th root. For square roots, there is a proof in the book (proof of Theorem A.10 on pages 622–623), and the author remarks that an analogous argument works for \(n\)th roots.
For the purposes of the exercise, you should assume that the \(n\)th root of a positive real number exists, and you should interpret the symbol \(\sqrt[n]{a}\) as meaning the unique positive \(n\)th root of \(a\) when \(a>0\).
The author claims that this exercise is used in Section 2.3, but I could not discover where.

Page 30

Notice that the converse of Remark 1.31 is false. For instance, if \(f(x)=x^3\), then \(f\) is one-to-one on the whole real line, but the derivative equals \(0\) when \(x=0\).

Page 31, Example 1.32

Notice that the range of \(f\) is all of R.

Page 32

There is a typographical error in the paragraph following the Warning. The three instances of \(\{1\}\) should all be \(\{0\}\).

Page 34, Exercise 1.5.0

In part a, delete the word “of”.

There is a typographical error in part d: the parentheses are mismatched. The left-hand side of the formula should be \( f^{-1}(f(\{0\})).\)

Page 35, Exercise 1.5.7

For “is used in several times” read “is used several times”.

Page 40

In Exercise 1.6.6, part a should include the initial sentence beginning “Suppose”.

Harold P. Boas