Analysis I

Corrected third edition

123 corrections · 3 awaiting a page number

1. p. 1On the final line, $-2$ should be in math mode.
2. p. 7In Example 1.2.6, Theorem 19.5.1 should be “Theorem 7.5.1 of Analysis II”.
3. p. 8In Example 1.2.7, “Exercise 13.2.9” should be “Exercise 2.2.9 of Analysis II”. In Example 1.2.8, “Proposition 14.3.3” should be “Proposition 3.3.3 of Analysis II”. In Example 1.2.9, “Theorem 14.6.1” should be “Theorem 3.6.1 of Analysis II”.
4. p. 9In Example 1.2.10, “Theorem 14.7.1” should be “Theorem 3.7.1 of Analysis II”.
5. p. 11In the final line, the comma before “For instance” should be a period.
6. p. 14“without even aware” should be “without even being aware”.
7. p. 17In Definition 2.1.3, add “This convention is actually an oversimplification. To see how to properly merge the usual decimal notation for numbers with the natural numbers given by the Peano axioms, see Appendix B.”
8. p. 19After Proposition 2.1.8: “Axioms 2.1 and 2.2” should be “Axioms 2.3 and 2.4”.
9. p. 20In the proof of Proposition 2.1.11, the period should be inside the parentheses in both parentheticals. Also, Proposition 2.1.11 should more accurately be called Proposition Template 2.1.11.
10. p. 23first paragraph: delete a right parenthesis in $f_3(a_3))$.
11. p. 27In the final sentence of Definition 2.2.7, the period should be inside the parentheses. In proposition 2.2.8, “$a$ is positive” should be “$a$ is a positive natural number”.
12. p. 29Add Exercise 2.2.7: “Let $n$ be a natural number, and let $P(m)$ be a property pertaining to the natural numbers such that whenever $P(m)$ is true, $P(m+\!+)$ is true. Show that if $P(n)$ is true, then $P(m)$ is true for all $m \geq n$. This principle is sometimes referred to as the principle of induction starting from the base case $n$“.
13. p. 31“Euclidean algorithm” should be “Euclid’s division lemma”.
14. p. 39in the sentence before Proposition 3.1.18, the word Proposition should not be capitalised.
15. p. 41In the paragraph after Example 3.1.22, the final right parenthesis should be deleted.
16. p. 45at the end of the section, add “Formally, one can refer to ${\bf N}$ as “the set of natural numbers”, but we will often abbreviate this to “the natural numbers” for short. We will adopt similar abbreviations later in the text; for instance the set of integers ${\bf Z}$ will often be abbreviated to “the integers”.”
17. p. 47In “In $\Omega$ did contain itself, then by definition”, add “of $\Omega$“. After “On the other hand, if $\Omega$ did not contain itself,” add “then by definition of $P$“, and after “and hence”, add “by definition of $\Omega$“.
18. p. 48In the third to last sentence of Exercise 3.2.3, the period should be inside the parenthesis.
19. p. 49“unique object $f(x)$” should be “unique object $f(x) \in Y$“, and similarly “exactly one $y$” should be “exactly one $y \in {\bf N}$“.
20. p. 49+: change all occurrences of “range” to “codomain” (including in the index). Before Example 3.3.2, add the following paragraph: “Implicit in the above definition is the assumption that whenever one is given two sets $X, Y$ and a property $P$ obeying the vertical line test, one can form a function object. Strictly speaking, this assumption of the existence of the function as a mathematical object should be stated as an explicit axiom; however we will not do so here, as it turns out to be redundant. (More precisely, in view of Exercise 3.5.10 below, it is always possible to encode a function $f$ as an ordered triple $(X, Y, \{ (x,f(x)): x \in X \})$ consisting of the domain, codomain, and graph of the function, which gives a way to build functions as objects using the operations provided by the preceding axioms.)”
21. p. 51Replace the first sentence of Definition 3.3.7 with “Two functions $f: X \to Y$, $g: X' \to Y'$ are said to be equal if and only if they have the same domain and codomain (i.e., $X=X'$ and $Y=Y'$), and $f(x)=g(x)$ for <I>all</I> $x \in X$.” Then add afterwards: “According to this definition, two functions that have different domains or different codomains are, strictly speaking, distinct functions. However, when it is safe to do so without causing confusion, it is sometimes useful to “abuse notation” by identifying together functions of different domains or codomains if their values agree on their common domain of definition; this is analogous to the practice of “overloading” an operator in software engineering. See the discussion [in the errata] after Definition 9.4.1 for an instance of this.”
22. p. 52In Example 3.3.9, replace “an arbitrary set $X$” with “a given set $X$“. Similarly, in Exercise 3.3.3 on page 55, replace “the empty function” with “the empty function into a given set $X$“.
23. p. 56After Definition 3.4.1, replace “a challenge to the reader” with “an exercise to the reader”. In Definition 3.4.1, “$S$ is a set in $X$” should be “$S$ is a subset of $X$“.
24. p. 62Replace Remark 3.5.5 with “One can show that the Cartesian product $X \times Y$ is indeed a set; see Exercise 3.5.1.”
25. p. 65Split Exercise 3.5.1 into three parts. Part (a) encompasses the first definition of an ordered pair; part (b) encompasses the “additional challenge” of the second definition. Then add a part (c): “Show that regardless of the definition of ordered pair, the Cartesian product $X \times Y$ is a set. (Hint: first use the axiom of replacement to show that for any $x \in X$, the set $\{ (x,y): y \in Y \}$ is a set, then apply the axioms of replacement and union.)”. In Exercise 3.5.2, add the following comment: “(Technically, this construction of ordered $n$-tuple is not compatible with the construction of ordered pair in Exercise 3.5.1, but this does not cause a difficulty in practice; for instance, one can use the definition of an ordered $2$-tuple here to replace the construction in Exercise 3.5.1, or one can make a rather pedantic distinction between an ordered $2$-tuple and an ordered pair in one’s mathematical arguments.)”
26. p. 66In Exercise 3.5.3, replace “obey” with “are consistent with”, and at the end add “in the sense that if these axioms of equality are already assumed to hold for the individual components $x,y$ of an ordered pair $(x,y)$, then they hold for an ordered pair itself”. Similarly replace “This obeys” with “This is consistent with” in Definition 3.5.1 on page 62.
27. p. 67In Exercise 3.5.12, add “Let $X$ be an arbitrary set” after the first sentence, and let $f$ be a function from ${\mathbf N} \times X$ to $X$ rather than from ${\mathbf N} \times {\mathbf N}$ to ${\mathbf N}$; also $c$ should be an element of $X$ rather than a natural number. This generalisation will help for instance in establishing Exercise 3.5.13.
28. p. 68In the first paragraph, the period should be inside the parenthetical; similarly in Example 3.6.2.
29. p. 71The proof of Theorem 3.6.12 can be replaced by the following, after the first sentence: ” By Lemma 3.6.9, ${\bf N} \backslash \{0\}$ would then have cardinality $n-1$. But ${\bf N}$ has equal cardinality with ${\bf N} \backslash \{0\}$ (using $x \mapsto x+1$ as the bijection), hence $n = n-1$, which gives the desired contradiction. Then in Exercise 3.6.3, add “use this exercise to give an alternate proof of Theorem 3.6.12 that does not use Lemma 3.6.9.”.
30. p. 73In Exercise 3.6.8, add the hypothesis that $A$ is non-empty.
31. p. 77“negative times positive equals positive” should be “negative times positive equals negative”. Change “we call $-n$ a negative integer“, to “we call $n$ a positive integer and $-n$ a negative integer“.
32. p. 89In the first paragraph, insert “Note that when $n=1$, the definition of $x^{-1}$ provided by Definition 4.3.11 coincides with the reciprocal of $x$ defined previously, so there is no incompatibility of notation caused by this new definition.”
33. p. 94bottom: “see Exercise 12.4.8” should be “see Exercise 1.4.8 of Analysis II”.
34. p. 97In Example 5.1.10, “1-steady” should be “0.1-steady”, “0.1-steady” should be “0.01-steady”, and “0.01-steady” should be “0.001-steady”.
35. p. 104In the proof of Lemma 5.3.7, after the mention of 0-closeness, add “(where we extend the notion of $\varepsilon$-closeness to include $\varepsilon=0$ in the obvious fashion)”, and after Proposition 4.3.7, add “(extended to cover the 0-close case)”.
36. p. 113In the second paragraph of the proof of Proposition 5.4.8, add “Suppose that $x>y$” after the first sentence.
37. p. 122Before Lemma 5.6.6: “$n^{th}$ root” should be $n^{th}$ roots”. In (e), add “Here $k$ ranges over the positive integers”, and after “decreasing”, add “(i.e., $x^{1/k} k$)”. One can also replace $x<1$ by $0 < x < 1$ for clarity.
38. p. 123near top: “is the following cancellation law” should be “is another proof of the cancellation law from Proposition 4.3.12(c) and Proposition 5.6.3”.
39. p. 124In Lemma 5.6.9, add “(f) $(xy)^q = x^q y^q$.”
40. p. ?In Exercise 5.6.5, replace “positive rational $q$ with $q > 1$” with “rational $q$ with $q > 0$”, and at the end of the exercise, ask what happens if $q<0$ or $q=0$ (rather than $q<1$).
41. p. 130Before Corollary 6.1.17, “we see have” should be “we have”.
42. p. 131In Exercise 6.1.6, $LIM$ should be $\mathrm{LIM}$.
43. p. 134In the paragraph after Definition 6.2.6, add right parenthesis after “greatest lower bound of $E$“.
44. p. 138In the second paragraph of Section 6.4, $-1$ should be in math mode (three instances). After $xL=L$ in the proof of Proposition 6.3.10, add “(here we use Exercise 6.1.3.)”.
45. p. 140In the first paragraph, $-1$ should be in math mode.
46. p. 143penultimate paragraph: add right parenthesis after “$L^+$ and $L^-$ are finite”.
47. p. 144In Remark 6.4.16, “allows to compute” should be “allows one to compute”.
48. p. 147“(see Chapter 1)” should be “(see Chapter 1 of Analysis II)”.
49. p. 148In the first sentence of Section 6.6, replace $(a_n)_{n=1}^\infty$ to $(a_n)_{n=m}^\infty$. After Definition 6.6.1, add “More generally, we say that $(b_n)_{n=m'}^\infty$ is a subsequence of $(a_n)_{n=m}^\infty$ if there exists a strictly increasing function $f: \{ n \in {\bf N}: n \geq m'\} \to \{ n \in {\bf N}: n \geq m\}$ such that $b_n = a_{f(n)}$ for all $n \in {\bf N}$.”.
50. p. ?In the hint for Exercise 6.6.5, “each natural numbers” should be “each natural number”.
51. p. 153Just before Proposition 6.7.3, “Section 6.7” should be “Section 5.6”.
52. p. 157At the end of Definition 7.1.6, add the sentence “In some cases we would like to define the sum $\sum_{x \in X} f(x)$ when $f: Y \to {\bf R}$ is defined on a larger set $Y$ than $X$. In such cases we use exactly the same definition as is given above.”
53. p. 161In Remark 7.1.12, change “the rule will fail” to “the rule may fail”.
54. p. 163In the proof of Corollary 7.1.14, the function $h$ should be replaced with its inverse (thus $h: Y \times X \to X \times Y$ is defined by $h(y,x) := (x,y)$. In Exercise 7.1.5, “Exercise 19.2.11” should be “Exercise 7.2.11 of Analysis II“.
55. p. 166In Remark 7.2.11 add “We caution however that in most other texts, the terminology “conditional convergence” is meant in this latter sense (that is, of a series that converges but does not converge absolutely).
56. p. 172In Corollary 7.3.7, $q$ can be taken to be a real number instead of rational, provided we mention Proposition 6.7.3 next to each mention of Lemma 5.6.9.
57. p. 175A space should be inserted before the (why?) before the first display.
58. p. 176In Exercise 7.4.1, add “What happens if we assume $f$ is merely one-to-one, rather than increasing?”. Add a new Exercise 7.4.2.: “Obtain an alternate proof of Proposition 7.4.3 using Proposition 7.4.1, Proposition 7.2.14, and expressing $a_n$ as the difference of $a_n+|a_n|$ and $|a_n|$. (This proof is due to Will Ballard.)”
59. p. 177In beginning of proof of Theorem 7.5.1, add “By Proposition 7.2.14(c), we may assume without loss of generality that $m \geq 1$ (in particulaar $|a_n|^{1/n}$ is well-defined for any $n \geq m$).”.
60. p. 178In the proof of Lemma 7.5.2, after selecting $N$, add “without loss of generality we may assume that $N \geq 1$“. (This is needed in order to take n^th roots later in the proof.) One can also replace $|a_n|^{1/n} \geq 1$ and $|a_n| \geq 1$ with $|a_n|^{1/n} > 1$ and $|a_n| > 1$ respectively.
61. p. 186In Exercise 8.1.4, Proposition 8.1.5 should be Corollary 8.1.6.
62. p. 187After Definition 8.2.1, the parenthetical “(and Proposition 3.6.4)” may be deleted.
63. p. 188In the final paragraph, after the invocation of Proposition 6.3.8, “convergent for each $m$” should be “convergent for each $n$“.
64. p. 189middle: in “Why? use induction”, “use” should be capitalised.
65. p. 190In the remark after Lemma 8.2.5, “countable set” should be “at most countable set”.
66. p. 193In Exercise 8.2.6, both summations $\sum_{m=N}^\infty$ should instead be $\sum_{m=0}^N$.
67. p. 198In Example 8.4.2, replace “the same set” with “essentially the same set (in the sense that there is a canonical bijection between the two sets)”.
68. p. 203In Definition 8.5.8, “every non-empty subset of $Y$ has a minimal element $\mathrm{min}(Y)$” should be “every non-empty subset $Z$ of $Y$ has a minimal element $\mathrm{min}(Z)$“.
69. p. 203In Proposition 8.5.10, “Prove that $P(n)$ is true” should be “Then $P(n)$ is true”.
70. p. 204Before “Let us define a special class….”, add “Henceforth we fix a single such strict upper bound function $s$“.
71. p. 205The assertion that $Y_\infty \cup \{ s(Y_\infty)\}$ is good requires more explanation. Replace “Thus this set is good, and must therefore be contained in $Y_\infty$” with : “We now claim that $Y_\infty \cup \{ s(Y_\infty)\}$ is good. By the preceding discussion, it suffices to show that $x = s( \{ y \in Y_\infty \cup \{ s(Y_\infty)\}: y < x \}$ when $x \in (Y_\infty \cup \{s(Y_\infty)\}) \backslash \{x_0\}$. If $x = s(Y_\infty)$ this is clear since $\{ y \in Y_\infty \cup \{ s(Y_\infty)\}: y < x \} = Y_\infty$ in this case. If instead $x \in Y_\infty$, then $x \in Y$ for some good $Y$. Then the set $\{ y \in Y_\infty \cup \{ s(Y_\infty)\}: y < x \}$ is equal to $\{ y \in Y: y < x \}$ (why? use the previous observation that every element of $Y' \backslash Y$ is an upper bound for $x$ for every good $Y'$), and the claim then follows since $Y$ is good. By definition of $Y_\infty$, we conclude that the good set $Y_\infty \cup \{ s(Y_\infty)\}$ is contained in $Y_\infty$“. In the statement of Lemma 8.5.15, add “non-empty” before “totally ordered subset”.
72. p. 206Remove the parenthetical “(also called the principle of transfinite induction)” (as well as the index reference), and in Exercise 8.5.15 use “Zorn’s lemma” in place of “principle of transfinite induction”. In Exercise 8.5.6, “every element of $x$” should be “every element of $X$“.
73. p. 208In Exercise 8.5.18, “Tthus” should be “Thus”. In Exercise 8.5.16, “total orderings of $P$” should be “total orderings of $X$“. In Exercise 8.5.19, the := should just be an =.
74. p. 211In Definition 9.1.1, the endpoints of an interval should only be defined when the interval is non-empty; similarly, in Examples 9.1.3, it is only the non-empty intervals with one or more endpoints infinite that should be called half-infinite or infinite. In Remark 9.1.2, add that for a non-empty interval $I$, the left endpoint can also be equivalently defined as $\inf I$ (why?), and similarly the right endpoint can be equivalently defined as $\sup I$. In particular, this makes it clear that these notions of endpoint are well-defined (two non-empty intervals that are equal as sets, will have the same endpoints).
75. p. 215Exercise 9.1.1 should be moved to be after Exercise 9.1.6, as the most natural proof of the former exercise uses the latter.
76. p. 216In Exercise 9.1.8, add the hypothesis that $I$ is non-empty. In Exercise 9.1.9, delete the hypothesis that $x$ be a real number.
77. p. 221At the end of Remark 9.3.7, $\lim_{x \in x_0; x \in E \backslash \{x_0\}}$ should be $\lim_{x \to x_0; x \in E \backslash \{x_0\}}$.
78. p. 222Replace the second sentence of proof of Proposition 9.3.14 by “Let $(a_n)_{n=0}^\infty$ be an arbitrary sequence of elements in $E$ that converges to $x_0$.”
79. p. 223Near bottom, in “Why? use induction”, “use” should be capitalised.
80. p. 224In Example 9.3.17, (why) should be (why?). In Example 9.3.16, “drop the set $X$” should be “drop the set $E$“, and change $\lim_{x \in x_0; x \in X}$ to $\lim_{x \to x_0; x \in X}$.
81. p. 225In Example 9.3.20, all occurrences of ${\bf R} - \{1\}$ should be ${\bf R} - \{-1\}$.
82. p. 226After Definition 9.4.1, add “We also extend these notions to functions $f: X \to Y$ that take values in a subset $Y$ of ${\bf R}$, by identifying such functions (by abuse of notation) with the function $\tilde f: X \to {\bf R}$ that agrees everywhere with $f$ (so $\tilde f(x) = f(x)$ for all $x \in X$) but where the codomain has been enlarged from $Y$ to ${\bf R}$.
83. p. 230In Exercise 9.4.1, “six equivalences” should be “six implications”. “Exercise 4.25.10” should be “Exercise 4.25.10 of Analysis II“.
84. p. 231In the second paragraph after Example 9.5.2, Proposition 9.4.7 should be 9.3.9. In Example 9.5.2, all occurrences of $x_0$ should be $0$. In the sentence starting “Similarly, if $(b_n)_{n=0}^\infty$…”, all occurrences of $a_n$ should be $b_n$.
85. p. 232In the proof of Proposition 9.5.3, in the parenthetical (Why? the reason…), “the” should be capitalised. Proposition 9.4.7 should be replaced by Definition 9.3.6 and Definition 9.3.3.
86. p. 233-234In Definition 9.6.1, replace “if” with “iff” in both occurrences.
87. p. 235In Definition 9.6.5, replace “Let …” with “Let $X$ be a subset of ${\bf R}$, and let …”.
88. p. 237Add Exercise 9.6.2: If $f,g: X \to {\bf R}$ are bounded functions, show that $f+g, f-g$, and $f \cdot g$ are also bounded functions. If we furthermore assume that $g(x) \neq 0$ for all $x \in X$, is it true that $f/g$ is bounded? Prove this or give a counterexample.”
89. p. 248Remark 9.9.17 is incorrect. The last sentence can be replaced with “Note in particular that Lemma 9.6.3 follows from combining Proposition 9.9.15 and Theorem 9.9.16.”
90. p. 252In the third display of Example 10.1.6, both occurrences of $(0,\infty)$ should be $(-\infty,0)$.
91. p. 253In the paragraph before Corollary 10.1.12, after “and the above definition”, add “, as well as the fact that a function is automatically continuous at every isolated point of its domain”.
92. p. 256In Exercise 10.1.1, $Y \subset X$ should be $Y \subseteq X$, and “also limit point” should be “also a limit point”.
93. p. 257In Definition 10.2.1, replace “Let …” with “Let $X$ be a subset of ${\bf R}$, and let …”. In Example 10.2.3, delete the final use of “local”. In Remark 10.2.5, $\subset$ should be $\subseteq$.
94. p. 259In Exercise 10.2.4, delete the reference to Corollary 10.1.12.
95. p. 260In Exercise 10.3.5, $X \subset {\bf R}$ should be $X \subseteq {\bf R}$.
96. p. 261In Lemma 10.4.1 and Theorem 10.4.2, add the hypotheses that $X, Y \subseteq {\bf R}$, and that $x_0,y_0$ are limit points of $X,Y$ respectively.
97. p. 262In the parenthetical ending in “$f^{-1}$ is a bijection”, a period should be added.
98. p. 263In Exercise 10.4.1(a), Proposition 9.8.3 can be replaced by Proposition 9.4.11.
99. p. 264In Proposition 10.5.2, the hypothesis that $f,g$ be differentiable on $[a,b]$ may be weakened to being continuous on $[a,b]$ and differentiable on $(a,b]$, with $g'$ only assumed to be non-zero on $(a,b]$ rather than $[a,b]$. In the second paragraph of the proof “converges to $x$” should be “converges to $a$“.
100. p. 265In Exercise 10.5.2, Exercise 1.2.12 should be Example 1.2.12.
101. p. 266“Riemann-Steiltjes” should be “Riemann-Stieltjes”.
102. p. 267In Definition 11.1.1, add “$X$ is nonempty and” before “the following property is true”, and delete the mention of the empty set in Example 11.1.3. In Lemma 11.1.4, impose the hypothesis that X be non-empty. (The reason for these changes is to be consistent with the notion of connectedness used in Analysis II and in other standard texts. -T.)
103. p. 276In the proof of Lemma 11.3.3, the final inequality should involve $f$ on the RHS rather than $g$.
104. p. 280In Remark 11.4.2, add “We also observe from Theorem 11.4.1(h) and Remark 11.3.8 that if $f: [a,b] \to {\bf R}$ is Riemann integrable on a closed interval $[a,b]$, then $\int_{[a,b]} f = \int_{(a,b]} f = \int_{[a,b)} f = \int_{(a,b)} f$.
105. p. 282In Corollary 11.4.4, replace” $|f| = f_+ - f_-$” by “$|f|$, defined by $|f|(x) := |f(x)|$“, and add at the end “(To prove the last part, observe that $|f| = f_+ - f_-$.)”
106. p. 283In the penultimate display, $(\overline{f_+}-\underline{f_-}(x))$ should be $(\overline{f_+}-\underline{f_-})(x)$.
107. p. 284Exercise 11.4.2 should be moved to Section 11.5, since it uses Corollary 11.5.2.
108. p. 288In Exercise 11.5.1, (h) should be (g).
109. p. ?At the start of the proof of Proposition 11.6.1, add “We may assume that $a \leq b$, since the claim is vacuously true otherwise.”.
110. p. 291In the paragraph before Definition 11.8.1, remove the sentences after “defined as follows”. In Definition 11.8.1, add the hypothesis that $\alpha$ be monotone increasing, and $X$ be an interval that is closed in the sense of Definition 9.1.15, and alter the definition of $\alpha[I]$ as follows. (i) If $I$ is empty, set $\alpha[I]:=0$. (ii) If $I = \{a\}$ is a point, set $\alpha[\{a\}] := \lim_{x \to a^+; x \in X} \alpha(x) - \lim_{x \to a^-; x \in X} \alpha(x)$, with the convention that $\lim_{x \to a^+; x \in X} \alpha(x)$ (resp. $\lim_{x \to a^-; x \in X} \alpha(x)$) is $\alpha(a)$ when $a$ is the right (resp. left) endpoint of $X$. (iii) If $I = (a,b)$, set $\alpha[(a,b)] := \lim_{x \to b^-; x \in X} \alpha(x) - \lim_{x \to a^+; x \in X} \alpha(x)$. (iv) If $I = [a,b)$, $(a,b]$, or $[a,b]$, set $\alpha[I]$ equal to $\alpha(\{a\}) + \alpha((a,b))$, $\alpha((a,b)) + \alpha(\{b\})$, or $\alpha(\{a\}) + \alpha((a,b)) + \alpha(\{b\})$ respectively. After the definition, note that in the special case when $\alpha$ is continuous, the definition of $\alpha[I]$ for $I = (a,b), [a,b), (a,b], [a,b]$ simplifies to $\alpha[I] = \alpha(b) - \alpha(a)$, and in this case one can extend the definition to functions $\alpha$ that are continuous but not necessarily monotone increasing. In Example 11.8.2, restrict the domain of $\alpha$ to $[0,+\infty)$, and delete the example of $\alpha[(-3,-2)]$.
111. p. 292In Example 11.8.6, restrict the domain of $\alpha$ to $[0,+\infty)$. In Lemma 11.8.4 and Definition 11.8.5, add the condition that $X$ be an interval that is closed, and $\alpha$ be monotone increasing or continuous.
112. p. 293After Example 11.8.7, delete the sentence “Up until now, our function… could have been arbitrary.”, and replace “defined on a domain” with “defined on an interval that is closed” (two occurrences).
113. p. 294The hint in Exercise 11.8.5 is no longer needed in view of other corrections and may be deleted.
114. p. 295In the proof of Theorem 11.9.1, after the penultimate display $|F(y)-F(x)| \leq M|x-y|$, one can replace the rest of the proof of continuity of $F$ with “This implies that $F$ is uniformly continuous (in fact it is Lipschitz continuous, see Exercise 10.2.6), hence continuous.”
115. p. 297In Definition 11.9.3, replace “all $x \in I$” with “all limit points $x$ of $I$“. In the proof of Theorem 11.9.4, insert at the beginning “The claim is trivial when $b=a$, so assume $b > a$, so in particular all points of $[a,b]$ are limit points.”. When invoking Lemma 11.8.4, add “(noting from Proposition 10.1.10 that $F$ is continuous)”.
116. p. 298After the assertion $F[J] = F(d)-F(c)$, add “Note that $F$, being differentiable, is continuous, so we may use the simplified formula for the $F$-length as opposed to the more complicated one in Definition 11.8.1.”
117. p. 299In Exercise 11.9.1, $q$ should lie in ${\bf Q} \cap (0,1)$ rather than ${\bf Q} \cap [0,1]$. In Exercise 11.9.3, $x_0$ should lie in $(a,b)$ rather than $[a,b]$. In the hint for Exercise 11.9.2, add “(or Proposition 10.3.3)” after “Corollary 10.2.9”.
118. p. 300In the proof of Theorem 11.10.2, Theorem 11.2.16(h) should be Theorem 11.4.1(h).
119. p. 310in the last line, “all logicallly equivalent” should be “all logically equivalent”.
120. p. 311In Exercise A.1.2, the period should be inside the parentheses.
121. p. 327In the proof of Proposition A.6.2, $0 \leq y \leq x$ may be improved to $0 < y < x$; similarly for the first line of page 328. Also, the “mean value theorem” may be given a reference as Corollary 10.2.9.
122. p. 329At the end of Appendix A.7, add “We will use the notation $X \equiv Y$ to indicate that a mathematical object $X$ is being identified with a mathematical object $Y$.”
123. p. 334In the last paragraph of the proof of Theorem B.1.4, “the number $a_n \dots a_0$ has only one decimal representation” should be “the number $a_n \dots a_1$ has only one decimal representation”.

Third edition (hardback)

66 corrections

1. General note: all references to “Analysis II” need to be renumbered to account for the new chapter numbering (basically, all chapter numbers need to be lowered by 11.)
2. p. 10footnote: “$f(0,0) := (0,0)$” should be $f(0,0) := 0$“.
3. p. 15In Section 2.1, “Guiseppe Peano” should be “Giuseppe Peano”.
4. p. 21In Remark 2.1.12, add the parenthetical comment “(augmented by adding a zero symbol $O$)” after the introduction of the Roman number system.
5. p. 29In the hint for Exercise 2.2.5, $n < m_0$ should be $n \leq m_0$.
6. p. 34“not all objects are sets” should be “it is not necessarily the case that all objects are sets”.
7. p. 35Definition 3.1.4 has to be given the status of an axiom (the axiom of extensionality) rather than a definition, changing all references to this definition accordingly. This requires some changes to the text discussing this definition. Firstly, in the preceding paragraph, “define the notion of equality” will now be “seek to capture the notion of equality”, and “formalize this as a definition” should be “formalize this as an axiom”. For the paragraph after Example 3.1.5, delete the first two sentences, and remove the word “Thus” from the third sentence. Exercise 3.1.1 is now trivial and can be deleted.
8. p. 37In Example 3.1.10, “so is singleton set” should be “the singleton set”; also, a right parenthesis is missing after (why?). In Axiom 3.4, “elements consists” should be “elements consist”.
9. p. 46In the first paragraph of Section 3.2, the appearances of the word “both” should be deleted.
10. p. 51In Remark 3.3.5, “the argument $f(x)$ of a function” should be “the argument of a function $f(x)$“. In Remark 3.3.6, “functions are not sets” should be “functions are not necessarily sets”, and similarly for “sets are not functions”. After “describes the function completely”, add “once the domain $X$ and range $Y$ are specified”. In Definition 3.3.7, add “two functions $f: X \to Y$ and $g: X' \to Y'$ are considered to be unequal if they have different domains $X \neq X'$ or different ranges $Y \neq Y'$ (or both)”.
11. p. 52The paragraph that “This notion of equality obeys the usual axioms (Exercise 3.3.1)” should be replaced by the following remark: “It is not immediately apparent that Definition 3.3.7 is compatible with the axioms of equality in Appendix A.7, although Exercise 3.3.1 below provides evidence towards this compatibility. There are at least three ways to address this issue. One is to regard Definition 3.3.7 as an axiom about equality of functions rather than a definition. Another is to provide a more explicit definition of a function in which Definition 3.3.7 becomes a theorem; for instance, one can define a function $f: X \to Y$ to be an ordered triple $(X,Y, G)$ consisting of a domain set $X$, a range set $Y$, and a graph $G = \{ (x,f(x)): x \in X\}$ that obeys the vertical line test, and use this latter graph to define the value of $f(x) \in Y$ for each element $x$ of the domain (cf. Exercise 3.5.10). A third way is to start with a mathematical universe ${\mathcal U}$ without any functions in it, and use Definition 3.3.7 to create a larger extension of this universe that contains function objects that behave as specified as in Definition 3.3.7. This final procedure however requires a bit more of the formalism of logic and model theory than is provided by this text, and so will not be detailed here.”
12. p. 54In Definition 3.3.17, the remark that a function is onto if $f(X)=Y$ should be moved to the next section, because the image $f(X)$ is not defined until that section.
13. p. 55In Example 3.3.22, “Axioms 2.2, 2.3, 2.4” should be “Lemma 2.2.10”. In Exercise 3.3.1, add “Of course, these statements are immediate from the axioms of equality in Appendix A.7 applied directly to the functions in question, but the point of the exercise is to show that they can also be established by instead applying the axioms of equality to elements of the domain and range of these functions, rather than to the functions itself.”.
14. p. 60A space missing between “the” and “Zermelo” in Remark 3.4.12.
15. p. 64The justification that the product set $\prod_{i=1}^n X_i$ given in Remark 3.5.8 is not quite correct if one is using the definition of an ordered n-tuple as defined in Exercise 3.5.2 (one has to restrict the range of the tuples to be surjective). As the correct version of this remark is part of Exercise 3.5.2, the second sentence of this remark should be replaced with a reference to that exercise. In Remark 3.5.10, $x_1,\dots,x_n$ should be $(x_1,\dots,x_n)$.
16. p. 66In Exercise 3.5.7, the direct sum $f \oplus g$ should be replaced by the pairing $(f,g)$.
17. p. 67In Exercise 3.5.12, $a_N(n++) = f(n,a(n))$ should be $a_N(n++) = f(n,a_N(n))$.
18. p. 68In Example 3.6.2, there is a superfluous period before the parenthetical (also the period after the parenthetical should be inside).
19. p. 70In the proof of Lemma 3.6.9, “Now define the function $g: X - \{x\}$ to $\{ i \in {\bf N}: 1 \leq i \leq n-1\}$” should be “Now define the function $g: X - \{x\} \to \{ i \in {\bf N}: 1 \leq i \leq n-1\}$” . In the 4th line of proof of Lemma 3.6.9: $1 \leq i \leq N$ should be $1 \leq i \leq n$.
20. p. 72In Exercise 3.6.8, the additional hypothesis that A is non-empty should be added. Also, the word “then” may be deleted.
21. p. 82In the footnote preceding Definition 4.2.1, add in the first sentence “… and $a$ is non-zero. Similarly, the identities $a/a = 1$ and $2*(a/a) = (2*a)/a$ cannot hold simultaneously if $0/0$ is defined.”
22. p. 94In the footnote, “Zahlen” is the German for “numbers”, not “number”.
23. p. 97In Definition 5.1.6 and Definition 5.1.8, $d(a_j,a_k)$ should be $|a_j-a_k|$ (for consistency with later definitions).
24. p. 103Near Proposition 5.3.3, “laws of equality” should be “axioms of equality”, and “law of substitution” should be “axiom of substitution”.
25. p. 104In the final line of the proof of Lemma 5.3.6, “eventually $\varepsilon$-close” should be “eventually $\varepsilon$-steady”.
26. p. 106In the first paragraph, “On obvious” should be “One obvious”.
27. p. 112In Definition 5.4.6, “if” should be “iff”.
28. p. 123Lemma 5.6.6(c) should read “$x^{1/n}$ is a non-negative real number, and is positive if and only if $x$ is positive”.
29. p. 124In the proof of Lemma 5.6.8, $(-a')/b$ should be $(-a')/b'$.
30. p. 135After Definition 6.2.6, add right parenthesis after “(also known as the greatest lower bound of $E$“.
31. p. 136In Definition 6.3.1, replace “sequence of real numbers” with “sequence of extended real numbers”.
32. p. 142Ellipsis is missing in the final display of Example 6.4.9.
33. p. 143In the penultimate paragraph, “$L^+$ and $L^-$ coincide” should be “$L^+$ and $L^-$ coincide and are finite”.
34. p. 144Below the proof of Proposition 6.4.12, a right parenthesis should be added after “(provided that $L^+$ and $L^-$ are finite”. Also, “(c) and (d)” should be “(d) and (e)”.
35. p. 150In Example 6.6.3, $0.001$ should be inserted between $1.001$ and $1.0001$.
36. p. 152In Exercise 6.6.3, add the following note: “To ensure the existence and uniqueness of the minimum, one either needs to invoke the well ordering principle (which we have placed in Proposition 8.1.4, but whose proof does not rely on any material not already presented), or the least upper bound principle (Theorem 5.5.9).” Similarly for Exercise 6.6.5.
37. p. 153In the proof of Lemma 6.7.1, the first equal sign in the display $d(x^{q_n}, x^{q_m}) = x^M (x^{q_n-q_m}-1) \leq \dots$ should be a $\leq$ sign.
38. p. 158In Example 7.1.7, $h(3)=c$ should be $h(3)=a$.
39. p. 160In Remark 7.1.10, all occurrences of $f(x)$ here should be $f(n)$.
40. p. 162In the third to last display, the small parenthesis near the end of the first term on the RHS should be moved to the outside (also, this pair of parentheses should be made larger).
41. p. 167In the proof of Proposition 7.2.12, “the sequence $(-1)^n a_n$” should be “the sequence $((-1)^n a_n)_{n=m}^\infty$“; similarly for “the sequence $S_n$” and “the sequence $a_N$“.
42. p. 174In the proof of Proposition 7.4.1, $(S_N)_{n=0}^\infty$ and $(T_M)_{m=0}^\infty$ should be $(S_N)_{N=0}^\infty$ and $(T_M)_{M=0}^\infty$ respectively.
43. p. 175In the first sentence, $m \leq N$ should be $m \leq M$.
44. p. 176In the proof of Proposition 7.4.3, “$\varepsilon$-close to $L$” should be $\varepsilon$-close to $L'$” in the last paragraph.
45. p. 188In the proof of Theorem 8.2.2, $X \subset {\bf N} \times {\bf N}$ should be $X \subseteq {\bf N} \times {\bf N}$. After definition 8.2.1, add “For finite sets $X$ we adopt the convention that series $\sum_{x \in X} f(x)$ are automatically considered to be absolutely convergent.”. “Taking suprema of this as $M \to \infty$” should be “Taking limits of this as $M \to \infty$“, and “by limit laws, and an induction on $N$” should be “by Exercise 7.1.5 and either Proposition 6.3.8 or Lemma 6.4.13”. In the preceding display, the first inequality should be an equality.
46. p. 189Before the final dusplay: “convergent for each $m$” should be “convergent for each $n$“.
47. p. 191In Lemma 8.2.3, $X$ should be assumed to be countable, rather than at most countable.
48. p. 193In Lemma 8.2.7, the last sentence should read “Then the series $\sum_{n \in A_+} a_n$ and $\sum_{n \in A_-} a_n$ are not absolutely convergent.”
49. p. 193Near the end of proof of Theorem 8.2.8, it would be (slightly) better to have $\lim_{j \to \infty} \sum_{0 \leq i \leq j} a_n$ rather than $\lim_{j \to \infty} \sum_{0 \leq i < j} a_n$.
50. p. 202In Exercise 8.4.3, “there exists an injection $f: A \to B$; in other words…” should be “there exists an injection $f: A \to B$ with $g \circ f: A \to A$ the identity map; in particular…”. (This is needed in order to establish the converse part of the question.)
51. p. 207In Exercise 8.5.6, $(x) \subset X$ should be $(x) \subseteq X$.
52. p. 209In Exercise 8.5.16, “$x,y \in P$” should be “$x,y \in X$“. In Exercise 8.5.18: A right parenthesis is missing after “… which contains $Y$“. “Tthus” should be “Thus”. In Exercise 8.5.20, $\Omega \subset 2^X$ should be $\Omega \subseteq 2^X$.
53. p. 212In Definition 9.1.1, “open intervals” should be “open interval”.
54. p. 216In Definition 9.1.22, $X \subset [-M,M]$ should be $X \subseteq [-M,M]$.
55. p. 217In Exercise 9.1.15, the hypothesis that $E$ is non-empty should be added.
56. p. 225In Example 9.3.17, “undefined (why)” should be “undefined (why?)”. Also, “in the textbook” should be “in some textbooks”. In Exampe 9.3.16, $\lim_{x \in x_0; x \in X}$ should be $\lim_{x \to x_0; x \in X}$.
57. p. 226In Example 9.3.21, all sequences here should start from $n=1$ rather than from $n=0$.
58. p. 230Exercise 11.25.10 should be Exercise 4.25.10 of Analysis II.
59. p. 237In Exercise 9.3.3, “Lemma 9.3.18” should be “Proposition 9.3.18”.
60. p. 257In Exercise 10.1.6, ${\bf R} - \{0\}$ should be ${\bf R} \backslash \{0\}$, and “differentiable on ${\bf R}$” should be “differentiable on ${\bf R} \backslash \{0\}$“. In Exercise 10.1.5, add “with the convention that $n x^{n-1} = 0$ when $n=0$“.
61. p. 264In Exercise 10.4.2(b), the limits should be over $(0,\infty) - \{-1\}$ rather than $(0,\infty)$.
62. p. 265In the proof of 10.5.2, “converges to $x$” should be “converges to $a$“.
63. p. 289In Exercise 11.6.5, add “For this exercise, you may use the second Fundamental Theorem of Calculus (Theorem 11.9.4); there is no circularity, because Corollary 11.6.5 is not used in the proof of that theorem.”
64. p. 295In the last paragraph of Section 11.8, a right parenthesis should be added at the end of the penultimate sentence.
65. p. 316In the proof of Proposition A.2.6, “$\sin(x)$ is increasing for $0 < x < \pi/2$” should be “$\sin(x)$ is increasing for $0 \leq x \leq \pi/2$“.
66. p. 330In Example A.7.3, “the substitution axiom” should read “the first form of the substitution axiom”. Then, at the end of the example, add “One can also obtain the conclusion $x=\sin(z^2)$ more directly by using the second form of the substitution axiom.”. At the end of the section, add “For most applications in analysis, one should not need to compare objects of different types: for instance, if $x$ is a set, and $y$ is a number, then one should not need to consider the question of whether $x=y$ is true or false. But for the purposes of doing set theory, it is convenient to adopt the convention that the statement $x=y$ is automatically false if $x,y$ are of different types; for instance, if one is treating natural numbers and vectors as objects of different types, then a natural number would not be equal to a vector. But sometimes we override this convention by identifying objects of one type with some objects of another type, e.g. when we identified natural numbers with their counterparts in the integers, or integers with their counterparts in the rationals, and so forth. This is technically an “abuse of notation”, but can be tolerated as long as one verifies that no violation of the axioms of equality occur by doing so.”

Second edition (hardback)

57 corrections

1. p. xiibottom: “solidifed” –> “solidified”.
2. p. xivtop: “to know how to to” –> “to know how to”.
3. p. 19In footnote 2, add: “In the converse direction, if we have $n=m$, then we may deduce $n++=m++$; this is the axiom of substitution (see Appendix A.7) applied to the operation $++$.”
4. p. 24after Definition 2.2.1: “defined $n+m$ for every integer $n$” should be “defined $n+m$ for every natural number $n$“.
5. p. 26after Proposition 2.2.6: “these notes” should be “this text”.
6. p. 28Proposition 2.2.14: “and Let” should be “and let”.
7. p. 30Lemma 2.3.3: “Natural numbers have no zero divisors” should read “Positive natural numbers have no zero divisors”.
8. p. 32Definition 2.3.11: Add the remark “In particular, we define $0^0$ to equal $1$.”
9. p. 37Example 3.1.10: “(why?)” should be “(why?))”.
10. p. 45“8-m, where n is a…” should be “8-m, where m is a…”. In Exercise 3.1.2, add Axiom 3.1 to the list of permitted axioms. In Exercise 3.1.1: (3.1.4) should be Definition 3.1.4.
11. p. 50In the first line, $h(2n+3)=h(2n+2)$ should be $h(2n+3)=2n+2$, and $N \backslash \{0\}$ should be ${\bf N} \backslash \{0\}$.
12. p. 55Exercise 3.3.1: $f \circ g$ and $\tilde f \circ \tilde g$ should be $g \circ f$ and $\tilde g \circ \tilde f$ respectively.
13. p. 59In Lemma 3.4.9, “Then the set … is a set” should read “Then there is a unique set of the form … . That is to say, there is a set $A$ such that for any $Y$, $Y \in A$ if and only if $Y$ is a subset of $X$.
14. p. 61In Exercise 3.4.8, Axiom 3.1 should be added to the list of permitted axioms.
15. p. 64In Example 3.5.9, “$(x_2,x_3) \in X_3$” should be “$(x_2,x_3) \in X_2 \times X_3$“.
16. p. 704th line of proof of Lemma 3.6.9: $1 \leq i \leq N$ should be $1 \leq i \leq n$. In the 6th line of proof of Proposition 3.6.8: Proposition 3.6.4 should be Lemma 3.6.9. After Lemma 3.6.9, add the following remark: “Strictly speaking, the expression $n-1$ has not yet been defined. For the purposes of this lemma, we temporarily define it to be the unique natural number $m$ such that $m++=n$ (which exists and is unique by Lemma 2.2.10).”
17. p. 81before Lemma 4.2.3: “product of a rational number” -> “product of two rational numbers”.
18. p. 84before Definition 4.2.6: a space is missing between “Proposition 4.2.4” and “allows”. Before this paragraph, add “In a similar spirit, we define subtraction on the rationals by the formula $x-y := x + (-y)$, just as we did with the integers.”
19. p. 86In Definition 4.3.2, “real numbers” should be “rational numbers”. In definition 4.3.4, “be a rational number” should be added after “Let $\varepsilon>0$“.
20. p. 88In Proposition 4.3.10(b), the hypothesis n>0 should be added.
21. p. 104proof of Lemma 5.3.7; after invoking Proposition 4.3.7, add “(extended in the obvious manner to the $\delta=0$ case)”.
22. p. 105after Proposition 5.3.10: $\lim_{n \to\infty} a_n$ should be $\hbox{LIM}_{n \to \infty} a_n$.
23. p. 108proof of Lemma 5.3.15: $n \geq N$ should be $n, m \geq N$. “This shows that $|a_n-a_m| \leq \varepsilon$” should read “This shows that $|a_n^{-1}-a_m^{-1}| \leq \varepsilon$“.
24. p. 115In the hint for Exercise 5.4.8, add “or Corollary 5.4.10” after “use Proposition 5.4.9”.
25. p. 120Add an additional exercise, Exercise 5.5.5: “Establish an analogue of Proposition 5.4.14, in which “rational” is replaced by “irrational”.”
26. p. 124Exercise 5.6.3: Add the hypothesis that x is non-zero (since the roots of 0 are not yet defined).
27. p. 126proof of Proposition 6.1.4: Proposition 5.4.14 should be Proposition 5.4.12.
28. p. 134In Definition 6.2.6(c) (and also on the first line of p. 135), $E - \{-\infty\}$ should be $E \backslash \{-\infty\}$.
29. p. 135Theorem 6.2.11(b), (c): Replace “Suppose that $M$” with “Suppose that $M \in {\Bbb R}^*$” (two occurrences). Exercise 6.2.2: Proposition 6.2.11 should be Theorem 6.2.11.
30. p. 144Cor. 6.4.14: line 4: ” .. for all $n \geq M$” should be ” .. for all $n \geq m$“
31. p. 146proof of Theorem 6.4.18: Replace “from Corollary 6.1.17” here by “from Lemma 5.1.15 (or more precisely, the extension of that lemma to the real numbers, which is proven in exactly the same fashion)”.
32. p. 151Exercise 6.6.5: Replace “the formula $n_j := \min\{n \in {\Bbb N}: |a_n-L| \leq 1/j\}$, explaining why the set $\{n \in {\Bbb N}: |a_n-L| \leq 1/j\}$ is non-empty” with “the recursive formula $n_j := \min\{n > n_{j-1}: |a_n-L| \leq 1/j\}$, with the convention $n_0=0$, explaining why the set $\{n > n_{j-1}: |a_n-L| \leq 1/j\}$ is non-empty”.
33. p. 164Definition 7.2.2: $(S_N)_{n=m}^\infty$ should be $(S_N)_{N=m}^\infty$.
34. p. 169Exercise 7.2.6: Add “How does the proposition change if we assume that $a_n$ does not converge to zero, but instead converges to some other real number $L$?”. After Corollary 7.3.2: “conditionally divergent” should be “not conditionally convergent”, similarly in Exercise 7.2.13.
35. p. 176“absolutely divergent series” should be “series that is not absolutely convergent”.
36. p. 177Theorem 7.5.1: “conditionally divergent” should be “not conditionally convergent”, and similarly “absolutely divergent” should be “not absolutely convergent”. Similarly for Corollary 7.5.3 on page 179.
37. p. 186Exercise 8.1.1: This exercise requires the axiom of choice, Axiom 8.1. In Exercise 8.1.4. $f(0), f(1), \ldots, f(n)$ should be $f(0), f(1),\ldots,f(n-1)$.
38. p. 192proof of Theorem 8.2.8: “absolutely divergent” should be “not absolutely convergent” (two occurrences).
39. p. 196Remark 8.3.6: “Paul Cohen (1934-)” should now be “Paul Cohen (1934-2007)”. :-(
40. p. 197Exercise 8.3.2: $f$ should be an injection rather than a bijection. In the definition of $g$, $\bigcup_{n=0}^\infty D_n$ should be $\bigcup_{n=1}^\infty D_n$ (two occurrences).
41. p. 200Exercise 8.4.1: $y \in y$ should be $y \in Y$.
42. p. 206Exercise 8.5.5: “$f(x) \leq_Y f(x')$” should be “$f(x) <_Y f(x')$ or $x=x'$“.
43. p. 208Exercise 8.5.19: $Y := \{y \in Y': y < x \}$ should be $Y = \{ y \in Y': y <' x \}$. In Exercise 8.5.20, the additional hypothesis “Assume that $\Omega$ does not contain the empty set $\emptyset$” should be added.
44. p. 214Lemma 9.1.21. One needs the additional hypothesis “We assume that $a<b$.”
45. p. 220Definition 9.3.6: “$f$ is $\varepsilon$-close to $L$ near $x_0$” should be “$f$, after restricting to $E$, is $\varepsilon$-close to $L$ near $x_0$“.
46. p. 228Proposition 9.4.7: change “three items” to “four items”, and add “(d): For every $\varepsilon > 0$, there exists a $\delta > 0$ such that $|f(x)-f(x_0)| \leq \varepsilon$ for all $x \in X$ with $|x-x_0| \leq \delta$.
47. p. 232proof of Proposition 9.5.3: after “Proposition 9.4.7”, add “(applied to the restriction of $f$ to the subdomain $X \cap (x_0,+\infty)$)”.
48. p. 252Proposition 10.1.7: One needs the additional hypothesis $x_0 \in X$. Similarly for Proposition 10.1.10, Theorem 10.1.13, and Proposition 10.3.1.
49. p. 253Definition 10.1.11: “For every $x_0 \in X$” should be “For every limit point $x_0 \in X$“.
50. p. 254Remark 10.1.14: Leibnitz should be Leibniz (two occurrences).
51. p. 256Exercise 10.1.1: “$x_0$ is also limit point of $Y$” should be “$x_0 \in Y$, and $x_0$ is also a limit point of $Y$“.
52. p. 257Definition 10.2.1: $x \in X$ should be $x_0 \in X$.
53. p. 262In the proof of Theorem 10.4.2,”$x_n = f^{-1}(y_0)$” should be “$x_n = f^{-1}(y_n)$“.
54. p. 271Remark 11.2.2: “constant on $f$” should be “constant on $E$“.
55. p. 290In Exercise 11.6.5, add “For this exercise, you may use the second Fundamental Theorem of Calculus (Theorem 11.9.4); there is no circularity, because Corollary 11.6.5 is not used in the proof of that theorem.”
56. p. 290In the proof of Proposition 11.7.1, in the third display, $[0,1]$ should be $|[0,1]|$.
57. p. 299In Exercise 11.9.1, the hint is misleading (it requires the mean value theorem for integrals rather than for derivatives, which is not covered in this text) and should be deleted.

First edition

27 corrections

1. p. 2item 3: “can you add” should be “Can you add”.
2. p. 9line 5: “right-hand side” should be “left-hand side”.
3. p. 10first display: $\frac{\partial^2}{\partial x \partial y}$ should be $\frac{\partial^2}{\partial y \partial x}$.
4. p. 5line 6 from bottom: $\sin(\pi/2-2)$ should be $\sin(\pi/2-z)$. (Actually, for pedagogical reasons, it may be slightly better to use $\pi/2+z$ throughout this example instead of $\pi/2-z$.)
5. p. 59Lemma 3.3.12: f should map Z to W, and h should map X to Y. In the proof of this lemma (on page 60): $g \circ h$ is a function from X to Z, and $f \circ g$ is a function from Y to W.
6. p. 67last paragraph: $\alpha \in A$ should be $\alpha \in I$.
7. p. 98In Exercise 4.2.1, Corollary 2.3.7 should be Corollary 4.1.9. In Exercise 4.2.6, $x,y,z$ should be rational numbers, not real.
8. p. 101In Definition 4.3.9, after “$x^0 := 1$“, add “; in particular, we define $0^0 := 1$“.
9. p. 127In Exercise 5.3.4: add “(Hint: use Exercise 5.2.2.)”.
10. p. 131line 12 from bottom: “they cannot be than” should be “they cannot be larger than”.
11. p. 175Exercise 6.6.3: In the hint, replace “introduce” by “recursively introduce”, and insert “; $n > n_{j-1}$” after “$|a_n| \geq j$” (two occurrences), with the parenthetical “(omitting the $n > n_{j-1}$ condition when $j=0$)” inserted after the recursive definition of $n_j$.
12. p. 181In Lemma 7.1.4(c), a period is missing at the end of $(\sum_{i=m}^n b_i)$.
13. p. 183In the proof of Proposition 7.1.8, $x$ should be replaced by $f(x)$ in every display of the proof in which it appears.
14. p. 197in second line of proof of Proposition 7.3.4: the second sum should be $\sum_{k=0}^\infty$ rather than $\sum_{k=0}^K$.
15. p. 216Exercise 8.1.9: It needs to be noted that this exercise requires the axiom of choice from Section 8.4.
16. p. 220Lemma 8.2.5: It needs to be noted that this lemma requires the axiom of choice from Section 8.4. Similarly, the case in Proposition 8.2.6 in which X is uncountable requires the axiom of choice also.
17. p. 227Exercise 8.3.2: $g(x) := f(x)$ should be $g(x) := f^{-1}(x)$.
18. p. 236last line: “for any good set Y'” should be “for any good set Y’ with $A \cap Y'$ non-empty”.
19. p. 250In Definition 9.10.3, “there exists an $M$” should be “there exists a real number $M$“. Also add “let $L$ be a real number” to the first sentence of the definition.
20. p. 255Proposition 9.3.9(b): $f(x_0)$ should be $L$.
21. p. 303Exercise 10.4.3(a): The limit should be in the set $(0,\infty) \backslash \{1\}$ rather than $(0,\infty)$.
22. p. 336line 13: replace “we have made no assumption on $\alpha$” with “the function $\alpha: {\Bbb R} \to {\Bbb R}$ could have been arbitrary”.
23. p. 337Exercise 11.8.1: Lemma 11.8.1 should be Lemma 11.8.4.
24. p. 337Exercise 11.8.5: In the last display, $f(0)$ should be $2f(0)$.
25. p. 342Exercise 11.9.1: “the function f is not differentiable” should be “the function $F(x) := \int_{[0,x]} f$ is not differentiable.
26. p. 383first display: $a_n \times \hbox{ten}^i$ should be $a_n \times \hbox{ten}^n$.
27. p. 387fourth display: $a_n$ should be $a_{n+1}$.