Analysis I

Last updated: May 13, 2023

Analysis, Volume I
Terence Tao
Hindustan Book Agency, January 2006. Fourth edition, 2022. (Also Springer, Fourth edition, 2002.)
Hardcover, 368 pages.

ISBN 81-85931-62-3 (first edition), 978-981-19-7261-4 (Springer fourth edition)

This is basically an expanded and cleaned up version of my lecture notes for Math 131A. In the US, it is available through the American Mathematical Society. It is part of a two-volume series; here is my page for Volume II. It is currently in its fourth edition. It will also be translated into French as “Le cours d’analyse de Terence Tao”.

There are no solution guides for this text.

Sample chapters (contents, natural numbers, set theory, integers and rationals, logic, decimal system, index)

Errata to older versions than the corrected third edition can be found here.

— Errata to the corrected third edition —

Page 1: On the final line, $-2$ should be in math mode.
Page 7: In Example 1.2.6, Theorem 19.5.1 should be “Theorem 7.5.1 of Analysis II”.
Page 8: In 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”.
Page 9: In Example 1.2.10, “Theorem 14.7.1” should be “Theorem 3.7.1 of Analysis II”.
Page 11: In the final line, the comma before “For instance” should be a period.
Page 14: “without even aware” should be “without even being aware”.
Page 17: In 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.”
Page 19: After Proposition 2.1.8: “Axioms 2.1 and 2.2” should be “Axioms 2.3 and 2.4”.
Page 20: In 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.
Page 23, first paragraph: delete a right parenthesis in $f_3(a_3))$ .
Page 27: In 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”.
Page 29: Add 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 $n$ “.
Page 31: “Euclidean algorithm” should be “Euclid’s division lemma”.
Page 39: in the sentence before Proposition 3.1.18, the word Proposition should not be capitalised.
Page 41: In the paragraph after Example 3.1.22, the final right parenthesis should be deleted.
Page 45: at 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”.”
Page 47: In “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 $\Omega$ “.
Page 48: In the third to last sentence of Exercise 3.2.3, the period should be inside the parenthesis.
Page 49: “unique object $f(x)$ ” should be “unique object $f(x) \in Y$ “, and similarly “exactly one $y$ ” should be “exactly one $y \in {\bf N}$ “.
Page 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.)”
Page 51: Replace 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.”
Page 52: In 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 $X$ “.
Page 56: After 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 “latex S$ is a subset of $X$ “.
Page 62: Replace Remark 3.5.5 with “One can show that the Cartesian product $X \times Y$ is indeed a set; see Exercise 3.5.1.”
Page 65: Split 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.)”
Page 66: In 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.
Page 67: In 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.
Page 68: In the first paragraph, the period should be inside the parenthetical; similarly in Example 3.6.2.
Page 71: The 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.”.
Page 73: In Exercise 3.6.8, add the hypothesis that $A$ is non-empty.
Page 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“.
Page 89: In 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.”
Page 94, bottom: “see Exercise 12.4.8” should be “see Exercise 1.4.8 of Analysis II”.
Page 97: In 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”.
Page 104: In 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)”.
Page 113: In the second paragraph of the proof of Proposition 5.4.8, add “Suppose that $x>y$ ” after the first sentence.
Page 122: Before 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., $x^{1/k} < x^{1/l}$ whenever $l>k$ )”. One can also replace $x<1$ by $0 < x < 1$ for clarity.
Page 123, near 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”.
Page 124: In Lemma 5.6.9, add “(f) $(xy)^q = x^q y^q$ .”
Page 130: Before Corollary 6.1.17, “we see have” should be “we have”.
Page 131: In Exercise 6.1.6, $LIM$ should be $\mathrm{LIM}$ .
Page 134: In the paragraph after Definition 6.2.6, add right parenthesis after “greatest lower bound of $E$ “.
Page 138: In 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.)”.
Page 140: In the first paragraph, $-1$ should be in math mode.
Page 143, penultimate paragraph: add right parenthesis after “ $L^+$ and $L^-$ are finite”.
Page 144: In Remark 6.4.16, “allows to compute” should be “allows one to compute”.
Page 147: “(see Chapter 1)” should be “(see Chapter 1 of Analysis II)”.
Page 148: In 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 $n \in {\bf N}$ .”.
Page 153: Just before Proposition 6.7.3, “Section 6.7” should be “Section 5.6”.
Page 157: At 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.”
Page 161: In Remark 7.1.12, change “the rule will fail” to “the rule may fail”.
Page 163: In 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“.
Page 166: In 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).
Page 172: In 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.
Page 175: A space should be inserted before the (why?) before the first display.
Page 176: In 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.)”
Page 177: In 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 $n \geq m$ ).”.
Page 178: In 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.
Page 186: In Exercise 8.1.4, Proposition 8.1.5 should be Corollary 8.1.6.
Page 187, After Definition 8.2.1, the parenthetical “(and Proposition 3.6.4)” may be deleted.
Page 188: In the final paragraph, after the invocation of Proposition 6.3.8, “convergent for each $m$ ” should be “convergent for each $n$ “.
Page 189, middle: in “Why? use induction”, “use” should be capitalised.
Page 190: In the remark after Lemma 8.2.5, “countable set” should be “at most countable set”.
Page 193: In Exercise 8.2.6, both summations $\sum_{m=N}^\infty$ should instead be $\sum_{m=0}^N$ .
Page 198: In 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)”.
Page 203: In 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 $\mathrm{min}(Z)$ “.
Page 203: In Proposition 8.5.10, “Prove that $P(n)$ is true” should be “Then $P(n)$ is true”.
Page 204: Before “Let us define a special class….”, add “Henceforth we fix a single such strict upper bound function $s$ “.
Page 205: The 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”.
Page 206: Remove 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 $X$ “.
Page 208: In Exercise 8.5.18, “Tthus” should be “Thus”. In Exercise 8.5.16, “total orderings of $P$ ” should be “total orderings of $X$ “.
Page 211: In 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).
Page 215: Exercise 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.
Page 216: In 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.
Page 221: At the end of Remark 9.3.7, $\lim_{x \in x_0; x \in E \backslash \{x_0\}}$ should be $\lim_{x \to x_0; x \in E \backslash \{x_0\}}$ .
Page 222: Replace 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 $x_0$ .”
Page 223: Near bottom, in “Why? use induction”, “use” should be capitalised.
Page 224: In 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 $\lim_{x \to x_0; x \in X}$ .
Page 225: In Example 9.3.20, all occurrences of ${\bf R} - \{1\}$ should be ${\bf R} - \{-1\}$ .
Page 226: After 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 ${\bf R}$ .
Page 230: In Exercise 9.4.1, “six equivalences” should be “six implications”. “Exercise 4.25.10” should be “Exercise 4.25.10 of Analysis II“.
Page 231: In 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 $b_n$ .
Page 232: In 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.
Page 233-234: In Definition 9.6.1, replace “if” with “iff” in both occurrences.
Page 235: In Definition 9.6.5, replace “Let …” with “Let $X$ be a subset of ${\bf R}$ , and let …”.
Page 237: Add 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.”
Page 248: Remark 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.”
Page 252: In the third display of Example 10.1.6, both occurrences of $(0,\infty)$ should be $(-\infty,0)$ .
Page 253: In 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”.
Page 256: In Exercise 10.1.1, $Y \subset X$ should be $Y \subseteq X$ , and “also limit point” should be “also a limit point”.
Page 257: In 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 $\subseteq$ .
Page 259: In Exercise 10.2.4, delete the reference to Corollary 10.1.12.
Page 260: In Exercise 10.3.5, $X \subset {\bf R}$ should be $X \subseteq {\bf R}$ .
Page 261: In 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.
Page 262. In the parenthetical ending in “$latex f^{-1} is a bijection”, a period should be added.
Page 263: In Exercise 10.4.1(a), Proposition 9.8.3 can be replaced by Proposition 9.4.11.
Page 264: In 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 $a$ “.
Page 265: In Exercise 10.5.2, Exercise 1.2.12 should be Example 1.2.12.
Page 266: “Riemann-Steiltjes” should be “Riemann-Stieltjes”.
Page 267: In 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, replace “connected” by “either connected or 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.)
In the start of Appendix A.1, “relations between them (addition, equality, differentiation, etc.)” should be “operations between them (addition, multiplication, differentiation, etc.) and relations between them (equality, inequality, etc.)”.
Page 276: In the proof of Lemma 11.3.3, the final inequality should involve $f$ on the RHS rather than $g$ .
Page 280: In 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 $\int_{[a,b]} f = \int_{(a,b]} f = \int_{[a,b)} f = \int_{(a,b)} f$ .
Page 282: In 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 $|f| = f_+ - f_-$ .)”
Page 283: In the penultimate display, $(\overline{f_+}-\underline{f_-}(x))$ should be $(\overline{f_+}-\underline{f_-})(x)$ .
Page 284: Exercise 11.4.2 should be moved to Section 11.5, since it uses Corollary 11.5.2.
Page 288: In Exercise 11.5.1, (h) should be (g).
Page 291: In 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 $\alpha[(-3,-2)]$ .
Page 292: In 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.
Page 293: After 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).
Page 294: The hint in Exercise 11.8.5 is no longer needed in view of other corrections and may be deleted.
Page 295: In 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.”
Page 297: In 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)”.
Page 298: After 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.”
Page 299: In 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”.
Page 300: In the proof of Theorem 11.10.2, Theorem 11.2.16(h) should be Theorem 11.4.1(h).
Page 310: in the last line, “all logicallly equivalent” should be “all logically equivalent”.
Page 311: In Exercise A.1.2, the period should be inside the parentheses.
Page 327: In 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.
Page 329: At 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 $Y$ .”
Page 334: In 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”.

— Errata to the fourth edition —

General: all instances of “supercede” should be “supersede”, and “maneuvre” should be “manoeuvre”.
Page 5: “theirbooks” should be “their books”.
Page 10: In Example 1.2.12, final paragraph, $4 x^{-2}$ should be $4 x^{-3}$ .
Page 15: “carry of digits” should be “carry digits”.
Page 16: The semicolon before $5 \times 3$ should be a colon.
Page 17: the computing language C should not be italicised.
Page 22: In Remark 2.1.5, the first “For instance” may be deleted.
Page 24: In the analysis of Case 1 of the proof of Theorem 1.5.8, $n \geq N$ should be $j \geq N$ (two occurrences).
Page 39: In the last part of Definiton 3.1.15, “if” should be “iff”.
Page ???: In the statement of Lemma 3.4.10, replace “Then the set” with “Then”.
Page 51: Delete the second part of Exercise 3.4.6 (it is redundant in view of Exercise 3.5.11), and replace “see also Exercise 3.5.11” with “see Exercise 3.5.11 for a converse to this exercise, which also helps explain why we refer to Axiom 3.11 as the “power set axiom”.”
Page 66: In Exercise 3.5.6, “the $A,B,C,D$ ” should be “the sets $A,B,C,D$ “.
Page 68: In Exercise 3.5.2, the domain of $a$ should be ${\bf N}$ rather than $X$ .
Page 69: In the paragraph before Definition 3.6.5, $N$ should be ${\bf N}$ .
Page 70: In the fifth line of the proof of Lemma 3.6.9, $1 \leq i \leq N$ should be $1 \leq i \leq n$ .
Page 87: In Proposition 4.3.7(b), add the following parenthetical: “Because of this equivalence, we will also use “ $x$ and $y$ are $\varepsilon$ -close” synonymously with either “ $x$ is $\varepsilon$ -close to $y$ ” or “ $y$ is $\varepsilon$ -close to $x$ “.
Page ???: Delete the last sentence of Exercise 4.4.3 (as the axiom of choice has not yet been formally introduced.)
Page 102: In the sixth line from the bottom of the proof, delete the first “yet”.
Page 104: In the statement of Lemma 5.3.6, delete the space before the close parenthesis.
Page 109: In the top paragraph (after Proposition 5.3.11), “On obvious guess” should be “One obvious guess”. In the proof of Lemma 5.3.15, after “ $|a_m - a_n| \leq c^2 \varepsilon$ “, “for all $n \geq N$ “, should be “for all $m,n \geq N$ “
Page 116: The paragraph after Remark 5.4.11 may be deleted, since it is essentially replicated near Definition 6.1.1.
Page 148: After Definition 6.6.1, “for all $n \in {\bf N}$ ” should be “for all $n \geq m'$ “.
Page 159: In Lemma 7.1.4(a), one can replace $m \leq n < p$ with $m \leq n \leq p$ .
Page 161: In the third display, on the right-hand side of the equation, the sizes of the first two left parentheses should be interchanged.
Page 174: In Proposition 7.4.1, $(S_N)_{n=0}^\infty$ should be $(S_N)_{N=0}^\infty$ , and similarly $(T_M)_{m=0}^\infty$ should be $(T_M)_{M=0}^\infty$ (two occurrences).
Page 177: In Exercise 7.3.2, add the requirement $x \neq 1$ to the geometric series formula.
Page 170: At the end of Section 8.2, add the following exercise (Exercise 8.2.7): “Let $f: {\bf N} \times {\bf N} \to {\bf R}$ be a function. Show that $\sum_{(n,m) \in {\bf N}^2} f(n,m)$ is absolutely convergent if and only if $\sum_{n=0}^\infty \sum_{m=0}^\infty |f(n,m)|$ is convergent.”
Page 198: In Example 8.4.2, replace “For any sets $I$ and $X$ ” with “For any set $X$ and non-empty set $I$ “.
Page 200: In Remark 8.3.5, “Exercise 7.2.6” should be “Exercise 7.2.6 of Analysis II“.
Page 201: In the first paragraph of Section 8.4, “Section 7.3” should be “Section 7.3 of
Analysis II“.
Page 205: In Proposition 8.5.10, replace $<_X$ with $<$ .
Page 207: In Exercise 8.5.15, replace the hint with “Apply Zorn’s lemma to the set of pairs $(X, \iota)$ , where $X$ is a subset of $B$ and $\iota: X \to A$ is an injection, after giving this set a suitable partial ordering.”
Page 215: In the final paragraph of the proof of Lemma 9.1.12, $(a-b)$ should be $(a,b)$ .
Page 219: In Remark 9.1.25, “Theorem 1.5.7” should be “Theorem 1.5.7 of Analysis II“.
Page 231: In Example 9.4.3, in the second limit, $x_0 \in x$ should be $x \to x_0$ .
Page 234: To improve the logical ordering, Proposition 9.4.13 (and the preceding paragraph) can be moved to before Proposition 9.4.10 (and similarly Exercise 9.4.5 should be moved to before Exercise 9.4.3, 9.4.4).
Page 237: At the start of Section 9.7, “a continuous function attains” should be “a continuous function on a closed interval attains”.
Page 249: In the first display of the proof of Theorem 9.9.16, $n \in N$ should be $n \in {\bf N}$ .
Page 252: In Definition 11.8.1, “a interval” should be “an interval”. In (ii), “ $X$ is the right endpoint” should be “ $a$ is the right endpoint”, and the final right parenthesis should be deleted. In (iii), $b^+$ should be $a^+$ , and the colons in the limits should be semicolons. In (iv), all occurrences of $\alpha(\dots)$ should be $\alpha[\dots]$ for notational consistency. Add an exercise to the effect that the limits in (ii), (iii) are well-defined, and refer to this exercise in the definition.
Page 253: In Lemma 11.8.4, the word “which” may be omitted.
Page 255: In Theorem 10.1.13(h), enlarge the parentheses around $\frac{f}{g}$ .
Page 271: In Remark 11.1.2, “Section 2.4” should be “Section 2.4 of Analysis II“.
Page 278: In Remark 11.3.5, replace “this is the purpose of the next section” with “see Proposition 11.3.12”. (Also one can mention that this definition of the Riemann integral is also known as the Darboux integral.)
Page 281: In Remark 11.3.8, “Chapter 8” should be “Chapter 8 of Analysis II“.
Page 292: In Remark 11.7.2, “Chapter 8” should be “Chapter 8 of Analysis II“.
Page 294: $\alpha(I)$ should be $\alpha[I]$ (two occurrences).
Page 297: In Definition 11.9.3, “all limit points $x$ of $I$ ” should be “all limit points $x$ of $I$ that are contained in $I$ “.
Page 314: In Examples A.2.1, “no conclusion on $x$ or $x^2$ ” should be “no conclusion for $x^2$ “.
Page ???: In Exercise B.2.3, “ $x$ is a terminating decimal” should be “ $x$ is a non-zero terminating decimal”, and “ $x$ is not at terminating decimal” should be “ $x$ is not a non-zero terminating decimal”.
General LaTeX issues: Use \text instead of \hbox for subscripted text. Some numbers (such as 0) are not properly placed in math mode in certain places. Some instances of \ldots should be \dots. \lim \sup should be \limsup, and similarly for \lim \inf.

Thanks to aaron1110, Adam, James Ameril, Paulo Argolo, William Barnett, José Antonio Lara Benítez, Dingjun Bian, Philip Blagoveschensky, Tai-Danae Bradley, Brian, Eduardo Buscicchio, Maurav Chandan, Matheus Silva Costa, Gonzales Castillo Cristhian, Ck, William Deng, Kevin Doran, Lorenzo Dragani, Evangelos Georgiadis, Elie Goudout, Ti Gong, Cyao Gramm, Christian Gz., Ulrich Groh, Yaver Gulusoy, Minyoung Jeong, Erik Koelink, Brett Lane, David Latorre, Kyuil Lee, Matthis Lehmkühler, Bin Li, Percy Li, Matthew, Ming Li, Mufei Li, Yingyuan Li, Manoranjan Majji, Mercedes Mata, Simon Mayer, Pieter Naaijkens, Vineet Nair, Cristina Pereyra, Huaying Qiu, David Radnell, Tim Reijnders, Issa Rice, Eric Rodriguez, Pieter Roffelsen, Luke Rogers, Feras Saad, Gabriel Salmerón, Vijay Sarthak, Leopold Schlicht, Marc Schoolderman, Rainer aus dem Spring, SkysubO, Tim Smith, Sundar, suinwethilo, Karim Taha, Chaitanya Tappu, Winston Tsai, Kent Van Vels, Andrew Verras, Daan Wanrooy, John Waters, Yandong Xiao, Hongjiang Ye, Luqing Ye, Christopher Yeh, Muhammad Atif Zaheer, and the students of Math 401/501 and Math 402/502 at the University of New Mexico for corrections.

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5 April, 2022 at 8:16 am

Mat

Does the implication from (b) to (a) in Proposition 9.3.9 require the axiom of choice? Let $B_\epsilon(a)$ be an open ball centered around $a$ with radius $\epsilon$ , we want to show for all $\epsilon$ there exists $\delta$ such that $f(B_\delta(x_0)\cap E)\subseteq B_\epsilon(L)$ given that all sequences $(a_n)_{n\in\mathbb{N}}$ in $E$ which converges to $x_0$ , we have $(f(a_n))_n$ converges to $L$ . Suppose towards a contradiction there exists $\epsilon$ such that all $B_\delta(x_0)$ contains a point $a\in B_\delta(x_0)$ where $d(f(a), L) > \epsilon$ . Then we can let $\delta=1/n$ for all natural numbers, and apply the axiom of choice to construct a sequence $(a_n)_n$ which converges to $x_0$ but $\lim f(a_n) \neq L$. The problem here is I have no nice way to describe the elements in $\{a\in B_\delta(x_0) : d(f(a), L) > \epsilon\}$ . The only thing I know about this set is that it is not empty.

5 April, 2022 at 3:05 pm

Terence Tao

Yes, (a weak form of) the axiom of choice is needed here.

12 April, 2022 at 6:52 pm

Anonymous

Professor, can you prove implication from (b) to (a) in Proposition 9.3.9 directly? not using reductio ad absurdum.

13 April, 2022 at 9:52 am

Terence Tao

I doubt it; more formally, I suspect that this direction of the implication (sequential continuity implies continuity) is not provable in intuitionistic logic, but I am not an expert in these matters.

13 April, 2022 at 10:39 am

Anonymous

Dr. Tao. Could you comment on the difference between your “Analysis I” and another book “An epsilon of room I: real analysis”? And which one is better for learning?

13 April, 2022 at 6:10 pm

Tim Smith

The second sentence of Lemma 3.4.8 is “Then the set {…} is a set.” The way it is worded, it seems to presuppose what it is trying to state. Maybe remove “the set” and just say “Then {…} is a set.”?

[Erratum added – T.]

5 May, 2022 at 1:51 am

Matthew

Dear Professor Tao,
Thanks for reading my comment! I have some questions below.
1. You mentioned in an earlier comment that in any mathematical theory with types or sorts (such as “natural number”), the property of being any specific sort is automatically a predicate, and that the statement “x=y” is automatically false if one is treating x and y as objects of different types. I am confused about the question of whether 1 and +∞ are objects of the same type(since they are both extended real numbers) or objects of different types(since 1 is a real number and +∞ is not). A similar question is, are 1 and 1/2 objects of the same type (since they are both real numbers) or objects of different types(1 is a natural number but 1/2 is not)?
2. I’ve noticed that there are not any axioms concerning the extended real numbers in the book. Should the existence of +∞ and -∞ be an axiom? Or should we define ±∞ set-theoretically? I find it quite strange to just add these two symbols to the set of real numbers.
3. You extend the operation of negation to R* by defining -(+∞):=-∞. I’m curious whether we should verify the axiom of substitution here. To be precise, I don’t know how to verify the statement “if x=+∞, then -x=-(+∞)” from this definition. Do you think it’s better to define this operation of negation by the sentence “if x=+∞，then -x:=-(+∞)” ?
Sorry for the language barrier. Thanks again for your patience!
Best regards,
Matthew

9 May, 2022 at 1:07 pm

Terence Tao

If one were to strictly adhere to type conventions (analogous to using a strongly typed language in software engineering) then one would have to distinguish between “the natural number 1”, “the integer 1”, “the real number 1”, and the “extended real number 1”, and these are all technically of different types and thus not necessarily comparable with each other. However, in practice we “abuse notation” and identify these objects together, and when asking for instance whether two numbers are equal one implicity applies the appropriate type conversion operator to place them to be of the same type. I talk about this also in this previous comment.

To introduce the extended reals, the only thing required of the new objects $+\infty$ and $-\infty$ is that they be distinct from each other and from all of the real numbers. The fact that we can do this follows for instance from Exercise 3.2.2.

One could define negation in long-hand notation “if $x=+\infty$ , then $-x := -\infty$ ” as you indicate, but it’s quicker to just use the shorthand “ $-(+\infty) := -\infty$ “. Again, this is technically an abuse of notation, but it does save space and does not cause much confusion in practice.

13 May, 2022 at 2:34 am

Matthew

Dear Professor Tao,
Thanks so much for your reply. I am reading the 3rd edition of Analysis Ⅰ and I found that Exercise 3.2.2 is about the proof of “A∉B or B∉A for any sets A,B” using the axiom of regularity. I don’t quite understand the reason why you indicated that the fact that we can make +∞ and -∞ distinct from each other and from all of the real numbers follows from Exercise 3.2.2.
What troubles me most in the book is the notion of equality. It is in your book that I first got to know the serious definition of equality and related axioms. I used to think that “x=y” means that “x and y are precisely the same thing”. But now it seems to me that the statement “x and y are the same object” is actually defined by the statement “x=y”. And the relation of equality, to a large extent, is just a matter of definition. We may define the relation of equality for a class of objects freely so long as the axioms of equality are all satisfied. Is that correct?
What troubles me most is the axiom of substitution, though. You said in a previous comment that any operation on objects defined using previous well-defined operations on those objects is also well-defined. Sometimes I succeed in understanding this (For instance, subtraction on the real numbers is well-defined, because subtraction is defined in terms of previous well-defined operations, namely addition and negation), but other times, I fail to understand this. For instance, suppose that m and m’ are integers such that m=m’, and suppose that (xn)_m is a sequence of reals with starting index m (I’m sorry for not typing this properly),and (xn)_m’ is a sequence of reals with starting index m’. Then is sup(xn)_m=sup(xn)_m’ ? I know that it’s super easy to prove, but my question is, do we actually need to prove this ?(Think about the axiom of substitution for subtraction.Though we can prove it step by step, we don’t need to.) Is it meaningful to think of these kinds of questions? (It seems I have been trapped by these questions.) I find that in most textbooks on analysis or algebra, the authors never talk about the axiom of substitution.
Thanks again for your patience.And I apologize for the language barrier.
Best Regards,
Matthew

14 May, 2022 at 10:02 am

Terence Tao

From Exercise 3.2.2, for every set $A$ there exists an object that does not lie in $A$ , namely $A$ itself. So, given the set of reals ${\bf R}$ , one can find an object (which one can call $+\infty$ by definition) which does not lie in ${\bf R}$ , and then one can find another object (which one can call $-\infty$ by definition) which does not lie in ${\bf R} \cup \{+\infty\}$ . Fo instance one can take $+\infty = {\bf R}$ and $-\infty = {\bf R} \cup \{+\infty\}$ , although this is a somewhat arbitrary choice.

Regarding the foundations of equality, perhaps it is worth making a distinction between the orthodox and “minimalist” foundations of first-order logic that one would see in a logic course, and the less formal and more “high-level” foundations used here which do not strictly adhere to these orthodox foundations but which conform more closely to the actual informal reasoning used by mathematicians in fields such as analysis. (This is somewhat similar to the distinction between low-level computer languages such as Assembly, and the high-level computer languages such as Python, Java, etc., that most programmers use in practice.) In the orthodox foundations, the universe of objects is fixed in advance, with equality given as a primitive notion. The axioms of equality are then postulates. This is a precise logical framework, but if one then wants to enlarge the universe by defining new objects one has to introduce some metamathematical constructions, in particular the notion of a conservative extension. This can be done, but requires a large digression into logic which is not really suitable for an introductory analysis text. Because of this, I have adopted a slightly less informal, “dynamic” approach which I have also described in this previous comment. Here, the universe of objects is not static, but is permitted to evolve by the introduction of classes of new objects, which can be formed for instance by taking class of existing objects and declaring some of them to be equal. One can then also introduce new operations on these new objects, as long as they respect the law of substitution, which is already established for the old objects and old operations but has to be verified for the new objects and operations. If one wants to do this formally, one can define the new objects more formally as equivalence classes of old objects (and new operations as functions on sets of equivalence classes), or by establishing a metatheorem (such as the one permitting extensions by definitions) that permits one to conservatively extend the old universe to incorporate these new objects and operations, but again this creates a significant digression from the primary task of the text, which is to teach analysis, and I would refer you instead to a logic textbook if you are really interested in developing this metatheory on a rigorous level rather than an informal one.

18 May, 2022 at 3:16 am

Matthew

Dear Professor Tao,
I’m sorry for perhaps posting several duplicated comments earlier.
In the Errata, you restate Exercise 3.5.12 to establish a rigorous version of Proposition 2.1.6 (Recursive definitions). I assume that this principle of recursive definition should be applied when exponentiating a real by a natural number in a rigorous way. Here is what I think: Let x be a real number. In order to exponentiate x by a natural number, let f:N×R→R be the function such that f(n,y)=x·y for all (n,y)∈N×R. Then there exists a unique function a:N→R such that a(0)=1 and a(n+1)=f(n,a(n)) for all n∈N. Then we define $x^n:=a(n)$ for all n∈N.
If my understanding is correct, then I think the statement “if n=n’∈N, then $x^n=x^{n'}$ ” follows from that fact that a:N→R is a function( functions obey the axiom of substitution). To be precise, we have $x^n=a(n)=a(n')=x^{n'}$ . Is this correct? Furthermore, I do think that we should verify yet another substitution law: if x=x’∈R, then $x^n=x'^n$ for all n∈N. This can be verified using the principle of induction. And I do not think this proof is so obvious that it can be omitted. What confuses me is that neither of the two laws of substitution is mentioned in the text.
I appreciate everything you have done for us.
Best Regards,
Matthew

18 May, 2022 at 5:27 am

Mat

I may not be qualified to answer this question but I will try to give my thought on this problem. Someone will correct me if I made a mistake.

Indeed, given $n=n'\in\mathbb{N}$ , one can show $x^n = x^{n'}$ by induction. As you have noticed, using axiom of substitution should suffice, as we have asserted $n=n'$ . The situation changed slightly for real numbers though, actually, for integers and rationals as well. Sometimes, we abuse notations and say $2//4 = 1//2$ but formally they are different (think of them as a formal pairs (2,4) and (1,2)). To resolve this issue, instead of calling them equal we say that they are equivalent. Some people prefer defining rationals as the set of all equivalence classes and when they write $1/2$ what they mean is the equivalence class of 1//2, i.e. the set of all formal pairs which are equivalent to 1//2. In this sense, $1/2=2/4$ is an equality as set. Now, think about how one defines a function which +1 over the rational numbers. We write $[a//b]=\{c//d : ad = bc\}$ as the equivalence class of $a//b$ . We can then define a function $f([a//b]) = [(a+b)//b]$ on rationals, but this function may not be well-defined. Say $[a//b] = [c//d]$ then $f([a//b])$ and $f([c//d])$ may not be the same as we are using different representatives. A counter-example as to why $f([a//b])=f([c//d])$ is not trivially true would be a function $g:\mathbb{Z}_3\to \mathbb{Z}_2 : [n]\mapsto [n]$ then $[3]=[6]$ in $\mathbb{Z}_3$ but $g([3])=[3]=[1]\neq [0]=[6]=g([6])$ in $\mathbb{Z}_2$ (is there any simpler example?). Put it in another way, let $[q],[q']$ be rationals where $q\text{ and }q'$ are equivalent formal rationals, we need to prove $f([q])=[q+1] = [q'+1]=f([q'])$ . Of course, this is written in a somewhat formal way. We may write it in a less formal way, let $q=q'\in \mathbb{Q}$ show $q+1=q'+1$ . Now you see the problem. Going back to your original question, if $x=x'\in\mathbb{R}$ is a genuine equality then $x^n={x'}^n$ is a consequence of substitution provided that $x\mapsto x^n$ is well-defined (it is, see Proposition 5.3.10). But more likely, question like this is saying that $x,x'$ are equivalent Cauchy sequences of rationals/formal reals, show $x^n={x'}^n$ . Then this is not a direct consequence of substitution.

18 May, 2022 at 7:08 am

Matthew

Thanks for the reply. After reading your comment carefully, I think there are two points to clarify. And I apologize in advance for any possible misunderstanding caused by the language barrier.
1. I do not think that we are “abusing notation” when writing $2//4=1//2$ . As Terrence says on page 85, $a--b$ is the space of all pairs equivalent to the ordered pair $(a, b)$ . Similarly, $a//b$ is actually the space of all pairs equivalent to $(a,b)$ . In other words, &latex a//b $ is already the set of all equivalence classes.
2.You indicate that the fact that “x↦x^n” is well-defined follows from Proposition 5.3.10. But as I understand it, Proposition 5.3.10 merely claims that multiplication is well-defined, not exponentiation.
Best Regards
Matthew

18 May, 2022 at 8:32 am

Mat

My definition of rationals is different than Tao’s, perhaps that is the source of confusion. He defines $\mathbb{Q}=\{\text{expressions of the form }a//b\}=\{a//b : a,b\in \mathbb{Z}\text{ and }b\neq 0\}$ whereas mine is $\mathbb{Q}'=\{\text{equivalence classes }[a//b]\}=\{[a//b] : \text{ ditto }\}$ . If we use Tao’s definition of rationals then you can ignore my “well-defined” argument above, as the function is well-defined in the sense that for all $x$ there exists a unique $y$ such that $f(x)=y$ . Instead, we can introduce a new concept of “well-behavedness”, and say a function is “well-behaved” if equivalent inputs lead to equivalent outputs, i.e., if $x\text{ and }x'$ are equivalent then $f(x)\text{ and }f(x')$ are equivalent. Then we can check all functions regarding integers, rationals and reals are “well-behaved”.

Regarding the function $x\mapsto x^n$ , if $n$ is a natural numbers then $x^n$ is just multiplying $x$ n-times, in which case, we can apply induction.

21 May, 2022 at 1:14 pm

Terence Tao

The general principle here (alluded to at various parts of the text, e.g., after the proof of Proposition 4.1.6) is that operations that are defined (either recursively or non-recursively) in terms of more primitive operations that are already known to be well-defined, will again be well-defined. Formally, this is established by either the recursion theorem (for recursive definitions) or by extension by definition) (for non-recursive definitions), basically along the lines you have indicated, but for the purposes of this analysis text I prefer to just invoke the above general principle at an informal level instead.

15 July, 2022 at 5:11 am

Matthew

Thank you professor.
I am so confused about the philosophic connotation of “equality”, and there is not a thorough explanation of it provided in the text.
Some authors say that “ $x=y$ “means that $x$ names the same object as does $y$ . And you mentioned in a previous comment that we could posit a preexisting mathematical universe of mathematical objects. But this viewpoint is too mysterious and reminds me of Plato’s theory of ideas or forms.
Others say that the essence of mathematics is symbolism, and that equality means that symbols are equivalent in a specific context. For instance, if $x = y$ , then $f(x) = f(y)$ for all well-defined operations $f$ , and $P(x)$ and $P(y)$ are equivalent statements for any well-defined property $P$ . They say that we write “ $x=y$ ” to mean that the symbols $x$ and $y$ are not distinguishable. Taking this viewpoint, symbols that appear in mathematics do not necessarily represent any mysterious “mathematical objects”, because they are just symbols. But I’m curious how we should use the phrase “exactly one” or “unique”? For instance, if there were two sets $\emptyset$ and $\emptyset'$ which were both empty, then we can prove that they would be equal to each other. And then you write that “ there can only be one empty set”. The symbols $\emptyset$ and $\emptyset'$ are equal, which means that anything that is true of $\emptyset$ is also true of $\emptyset'$ . But why are they somehow unified into just one symbol $\emptyset$ ? Is this because these two symbols cannot be distinguished in a specific context (namely, set theory)?
I wish you could explain it further, and I truly appreciate everything you have done for us.

15 July, 2022 at 5:19 am

Anonymous

The point is not define equal but unequal as primitive

Ft the intuitionist

17 July, 2022 at 8:16 am

Terence Tao

In mathematical logic, equality is taken as a primitive relation; it is agnostic as to what equality actually “means” – this is a question for the philosophers rather than for the mathematicians – but instead focuses on the axioms that equality is required to satisfy, such as the axiom of substitution. So you are free to interpret the notion of equality as you wish, so long as it is consistent with the axioms (cf. Remark 2.1.14 of my text). Other mathematical concepts, such as uniqueness, can then be defined in terms of this primitive equality relation.

It may also help to keep the map-territory relation in mind – an object is not the same thing as the string of symbols used to reference that object. For instance, $1+1$ and $2$ are distinct strings of symbols, and thus different from each other on a syntactic level, but on a semantic level they represent the same mathematical object, which in the notation of my book is the number $(0+\!\! +)+\!\! +$ ; this identity of representation is of course formalized in mathematics by the familiar identity $1+1=2$ . In most of mathematics, semantic equality is far more important than syntactic equality (though the latter is important in some parts of mathematics, such as proof formalization, mathematical logic, or certain aspects of abstract algebra), and so it becomes the default notion of equality used throughout the subject. (It does occasionally happen though that confusion is caused due to a conflation of the two types of equality, for instance through the perennial debates on whether $0.999\dots$ is really “equal” to 1, or whether fractions such as $1/2$ and $3/6$ should be considered “equal” or merely “equivalent”. These sorts of confusions can usually be resolved by carefully distinguishing between an object and its formal representation(s), for instance making a distinction between a number and a numeral.)

17 July, 2022 at 10:36 am

Anonymous

Happy Birthday!

17 July, 2022 at 5:46 pm

Matthew

Sorry, professor, some formulas in my last comment didn’t work well.
After reading your comment, I find that you seem to interpret the notion of equality in the following way: there is a mathematical universe (whatever that really is) of mathematical objects. And for each object, we may use distinct strings of symbols as its formal representations. And the equation “ $x=y$ ”means that the symbols $x$ and $y$ represent the same object in the mathematical universe. Is my understanding correct?
The reason why I elaborate on the above philosophic connotation of equality is that I wish to resolve the following confusion: if there were, for instance, two sets $\emptyset$ and $\emptyset'$ which were both empty, then by Definition 3.1.4, $\emptyset=\emptyset'$ . Then the empty set is unique and is denoted by $\emptyset$ . I wish to know the exact reason why we could unify the two symbols $\emptyset$ and $\emptyset'$ into just one symbol $\emptyset$ once we prove that $\emptyset=\emptyset'$ . I think this unification of symbols is only reasonable if we adopt the above interpretation for equality. However, if there were some other interpretation for“ $x=y$ ”which does not suggest that $x$ and $y$ represent the same object (you indicate in your last comment that we are free to interpret the notion of equality so long as the interpretation is consistent with the axioms), then maybe we would not be able to claim that there is exactly one empty set. This is because the equation $\emptyset=\emptyset'$ does not necessarily mean that $\emptyset$ and $\emptyset'$ represent the same object.

18 July, 2022 at 7:01 am

Terence Tao

By the axiom of substitution, once we have established that $\emptyset = \emptyset'$ , then the truth value of any sentence involving $\emptyset'$ is unchanged if we replace one or more of the instances of $\emptyset'$ with $\emptyset$ , and vice versa. So there is no substantive alteration to the theory if we decide to “retire” the symbol $\emptyset'$ by replacing all of its instances with $\emptyset$ . On the other hand, one is also free to keep the redunant symbol $\emptyset'$ if one wishes, much as we have different numeral representations (e.g., $II = 2 = (0+\!\!+)+\!\!+ = +2 = 2.0 = 2/1$ ) for the same number.

In modern first-order logic one makes a distinction between syntax (manipulation of a formal language of strings of symbols) and semantics (interpretation of these strings in a theory). A given theory – which is a syntactic construct – can be interpreted semantically by one or more structures (aka models), which assign meaning to each of the components of the theory, for instance assigning a specific mathematical object to each constant symbol. Within each such interpretation, two expressions $x, y$ would be considered equal (or more precisely, the sentence $x=y$ would be considered a true sentence) if they represent the same object in that structure. But if one prefers, one could take a purely formalist perspective and manipulate the theory on a purely syntactic level without reference to any interpretations (so in particular sentences such as $x=y$ become formal strings with no inherent meaning). Furthermore, if one is considering multiple structures, it is possible that equality of two expressions only holds in some structures and not in all. For instance, in the theory of rings (in which each ring becomes a structure of the theory), the sentence $1+1=0$ is true in some structures (e.g., the ring ${\bf Z}/2{\bf Z}$ ) but not in others (e.g., the ring ${\bf Z}$ ). In such cases it would become dangerous to syntactically identify $1+1$ and $0$ in the general theory of rings. However, if two expressions are equal in every interpretation, then it is quite safe to identify them syntactically (in part due to the Godel completeness theorem, which then assures us that the equality of these two expressions is in fact a theorem of the theory). For instance, in the theory of rings, the associative law of addition guarantees that $(x+y)+z = x+(y+z)$ for all $x,y,z$ , so one can safely identify these two expressions $(x+y)+z$ , $x+(y+z)$ (and one often introduces a third expression $x+y+z$ to be identified with the other two in order to reduce notational clutter).

In any event, these topics would be discussed far more extensively in a text on mathematical logic, rather than an analysis textbook. For instance I can recommend Enderton’s “A mathematical introduction to logic”, which was the text I first learned much of this material from.

27 May, 2022 at 5:37 am

In Section A.4. regarding “variables and quantifiers”, in the sentence:

“Let P(x) be some statement depending on a free variable x.”

is “P” also considered a (free? bound?) “variable”?

27 May, 2022 at 2:18 pm

Terence Tao

$P$ is a predicate. In second-order logic, one can view predicates themselves as variables that can be quantified over, but for beginners I would not recommend working with second-order logic until one is completely comfortable with first-order logic.

27 May, 2022 at 6:59 pm

At the end of the section “Universal quantifiers”, should the statement “ $\forall x\in X: P(x)$ is true” be understood as “P(x) is true for all x of type T” (with T=X) as written at the beginning of the section? Or “ $\in X$ ” is part of the whole predicate ( $\in X$ and P(x))?

[Either interpretation is acceptable at this level of formalization. (If one wants to formalize at the level of a computer verified proof then one can be slightly more careful with the syntax, but for ordinary mathematical reasoning it does not matter much.) -T]

12 June, 2022 at 10:51 am

Anonymous

Dr. Tao: The 3rd edition is a very good in terms of print quality, but the binding is terrible. One can see the thread and holes of the binding. This is not doing justice to the excellent book content. Maybe change the publisher next time, or ask them to amend this. People buy hard covers to keep it for long term, and it should be in excellent condition.

13 June, 2022 at 12:28 am

Jorge Silva

Dear Professor Tao, I have a suggestion, in light of corrections in Definition 11.8.1, to ensure that the $\alpha$ -lenght of a bounded interval $I$ is well-defined in the case when $\alpha$ is a monotone increasing function, one can show that if $f : X\rightarrow\mathbf{R}$ is a monotone function and if $x_0$ is an adherent point of $X\cup(x_0, \infty)$ then $f(x_0+)$ is finite, and the same for $f(x_0-)$ , this can perhaps be added as an exercise in section 9.8.

13 June, 2022 at 1:26 am

Jorge Silva

Oops, I forgot to add the hypotesis that $X$ is a closed subset of $\mathbf{R}$ .

13 June, 2022 at 2:07 pm

Anonymous

Dr. Tao:
Let $A,B$ be sets in $X, Y$ . For example, $A=B=[0,1]$ , and $X=Y=R$ . By definition:

$A\times B = \{(x,y)\in X\times Y|x\in A\wedge y\in B\}$

By De Morgan’s law,

$(A\times B)^c = \{(x,y)\in X\times Y|x\in A^c\vee y\in B^c\} = (A^c\times Y)\cup(X\times B^c)\cup(A^c\times B^c)$

Is the last term $(A^c\times B^c)$ necessary? I tried to draw a picture using the example, and the last term doesn’t contribute anything new. Since we are taking the union, and the first two terms already include it (since $A^c\in X$ and $B^c\in Y$ ). Could we just drop the last term in the $(A\times B)^c$ formula? It seems redundant.

[Yes. Note that there was no reason to include this additional set in the first place (see Axiom 3.4). -T]

18 June, 2022 at 2:35 pm

Anonymous

In Exercise B.2.3, to prove that $x$ has two different decimal representations if it is derminating decimal, $x$ must be different from zero, since zero is terminating decimal but its decimal representation is unique.

[Erratum added, thanks – T.]

15 July, 2022 at 8:58 pm

Anonymous

Dear Prof Tao.

In proposition 8.2.6(c), there are two parts.
when we are proving “Conversely, if h:X -> R…” part, Is there still premise which is X = X_1 union X_2 ?

[Yes, this is the premise for both directions of this part of the proposition (otherwise the second part would not make sense). -T.]

18 July, 2022 at 8:26 pm

Muhammad Atif Zaheer

Dear Prof,

In Theorem 10.4.2 (Inverse Function Theorem for a single variable) it is not alluded that $y_0 = f(x_0)$ is a limit point of $Y$ though it follows immediately from continuity of $f$ at $x_0$ .

19 July, 2022 at 9:33 am

Anonymous

Where did he say that $y_0=f(x_0)$ is a limit point of Y?

22 July, 2022 at 7:20 pm

Muhammad Atif Zaheer

He didn’t say that but in order for $f^{-1}$ to be differentiable at $y_0 = f(x_0)$ , $y_0$ should necessarily be a limit point of $Y$ .

31 August, 2022 at 4:21 pm

Anonymous

Hi Prof Tao. I’m following the 3rd Edition of Analysis I. In the (informal) Definition 3.1.1, it is stated that a set $A$ is any unordered collection of objects. Is anything lost if I omit “unordered”, that is, I simply say that a set $A$ is any collection of objects? Now, if I need to ascertain if $\{1,2,3\}$ equals $\{2,3,1\}$ , I refer to Definition 3.1.4, by which those two sets are equal. (I asked this here: https://math.stackexchange.com/questions/4522573/ where some said that “unordered” term is necessary in Definition 3.1.1.)

21 September, 2022 at 3:33 pm

smridhsharma

Hi Dr Tao. I had a question regarding exercise 6.3.4. At the end of the exercise it is said to find the flaw in example 1.2.3. My first guess was that the flaw was the very assumption that the limit exists. But after having given it some more thought, I think that the flaw might be something else. This is because, the hint for proving that $x^n$ diverges for $x>1$ was to prove by contradiction. So think that the assumption that $x^n$ converges in example 1.2.3 does lead to the contradiction that the limit $L=0$ . This contradicts the definition of convergence of a sequence. But if this is the flaw, it arose due to the assumption that $x^n$ converges. Is the flaw the assumption that the limit exists or is the flaw something else?

[Yes, the flaw is in assuming that the limit exists (which is not the case when $|x| > 1$ or $x=-1$ -T.]

23 September, 2022 at 6:42 am

lorenzo

Hello, has the fourth edition officially been published?
I’m asking because in this page you write “It is currently in its third edition (corrected), with a fourth edition in preparation.” but one can already find some copies of the fourth edition on Amazon so I was wondering whether these are legitimate copies of the book. Thanks.

[The information at the top of this page was out of date and is now updated; thanks for letting me know about the issue. -T]

26 October, 2022 at 11:59 pm

Matthew

Dear Professor Tao,
In Example 8.4.2, if we set $I:=\emptyset$ and $X\neq\emptyset$ , then the Cartesian product $\prod_{i\in I}X$ is the singleton set whose element is the empty function $f:\emptyset\rightarrow\emptyset$ , while $X^I$ is the singleton set whose element is a different empty function $g:\emptyset\rightarrow X$ . Thus strictly speaking, I think they are different sets.

[Subtle catch! Erratum added, thanks – T.]

29 October, 2022 at 7:50 pm

Anonymous

Dear professor Tao, I have a question. When defining the equality of integers, we proved four properties.(reflex, symmetry, transitive, substitution) But I didn’t do that in natural numbers. Why is that?

[The natural numbers were postulated to exist (and, thus, implicitly, postulated to already obey the axioms of equality). The integers, by constrast, were constructed from the natural numbers, without further postulation, and so equality axioms for the integers would also have to be verified. -T]

4 November, 2022 at 2:18 am

Matthew

I will try to give my thought on this question, and corrections are most welcome.
The natural numbers are considered as “primitive objects” (not defined in terms of other objects) in the text, and the four axioms of equality mentioned in the appendix are automatically true for those objects. For instance, if we know that $n=m$ for some natural numbers $n$ and $m$ , then the axiom of symmetry ensures us that $m=n$ . Similarly, the increment operation introduced in the Peano Axioms is considered a primitive operation, thus $n=m$ also implies that $n++=m++$ (this is the axiom of substitution applied to the operation $++$ ).
Sets are also considered as a primitive type of mathematical objects, and Definition 3.1.4 (Equality of sets) should be viewed as an axiom rather than a definition. We do not need a definition of equality for sets because those primitive objects are understood to have an equality relation assigned to them(equality is also taken as a primitive relation, and it is agnostic as to what equality actually means), and our definition for equality (namely, Definition 3.1.4) would probably lead to contradiction. Taking this viewpoint, one does not need to verify statements such as “Given any sets $A$ and $B$ , if $A=B$ , then $B=A$ “, since this is automatic from the laws of first-order logic. Instead, one should verify that the AXIOM of extensionality (which is Definition 3.1.4, but not viewed as a definition any more) is CONSISTENT with the axioms of equality. For instance, if we restated the axiom of extensionality as “Given any sets $A$ and $B$ , $A=B$ iff $\forall x\in A x\notin B$ “, then this would contradict the axiom of reflexivity.
However, equality amongst integers seems not to be taken as a primitive relation in the text, but rather a defined one. Thus statements such as “if $n=m$ , then $m=n$ ” are not automatic. If one wished to avoid this, one could use the set-theoretic machinery of equivalence classes as a substitute (as commented on page 75). But the author emphasizes the point that set theory is not necessary to construct the real number system.
These topics concerning mathematical logic are briefly discussed in some previous comments.

6 November, 2022 at 12:40 am

Anonymous

Dear professor Tao, I love your book. I think it’s the best among the books on analysis. Speaking of which, could you recommend an undergraduate abstract algebra book? The book you’re going to use if you’re going to teach algebra

7 November, 2022 at 10:14 am

Anonymous

He doesn’t teach algebra.

12 December, 2022 at 6:56 am

Anonymous

I know but.. he taught linear algebra

7 November, 2022 at 4:09 am

lorenzo

In the errata to the fourth edition the item: ” “Giuseppe” should be “Guiseppe”.” is wrong. “Guiseppe” in not a name in italian, “Giuseppe” is. “Giuseppe Peano” is thus the correct name (I am a native speaker).

[Removed, thanks – T.]

9 November, 2022 at 11:56 pm

suinwethilo

Dear Professor Tao,
I find some tricky points in Definition 9.1.1.
First, in order to define the left endpoint of an interval $I$ , do you think it’s worth excluding some weird possibility like $[1,3]$ may be equal to $[2,4]$ ? I know it’s quite easy to prove that $[1,3]\neq[2,4]$ but this fact does requires a proof, doesn’t it? Actually, there is indeed some “inconsistence of notation” if we take the empty set into consideration. For instance, we have $(1,-1)=[1,-1]=[2,0]=\emptyset$ . And we certainly can’t define the left endpoint of $(1,-1)$ to be $1$ and the left endpoint of $[2,0]$ to be 2 (which is the way left endpoints are defined in the text), since this would contradicts the axiom of substitution for equality.
I think the right way to define the left endpoint of an interval $I$ is by taking the left endpoint of $I$ to be just $inf(I)$ . We don’t have to worry about violating the axiom of substitution here, since the infimum operation is already known to be well-defined. Furthermore, one can prove that $inf[1,3]=1$ and then concludes that the left endpoint of $[1,3]$ is $1$ .
Same for right endpoints.
Another question is about the definition of a “bounded interval”. According to the definition given in the text, $[+\infty,-\infty]$ (which is exactly the empty set) is referred to as doubly-infinite and is therefore not a bounded interval. And in order for the notion of a bounded interval (the way it is defined in the text) to be well-defined, I think one should exclude some weird possibility like $(1,3)$ might be equal to $(-\infty, 2)$ . Of course they are distinct sets, but I think a proof is needed. And I think the right way is to claim that an interval $I$ (which is a subset of $\mathbb{R}^{*}$ ) is a bounded interval iff $I$ is contained in $\mathbb{R}$ and is bounded in the sense of Definition 9.1.22. Now we don’t have to worry about violating the axiom of substitution. Finally, if $I$ is a bounded interval and is non-empty, we can define the length of $I$ to be $sup(I)-inf(I)$ , and if $I$ is empty, the length is just $0$ . This could be a substitute for Definition 11.1.8 because one has good reasons to doubt that Definition 11.1.8 is well-defined.
Just some rough thought, and I apologise in advance if these seem senseless and inchoate.

[Erratum added to restrict notions of endpoint to non-empty intervals (so in particular the empty interval becomes bounded) -T.]

10 December, 2022 at 6:19 am

Anonymous

Dear Professor Tao,
I read a previous comment in which you talked about the subtle difference between a function and a well-formed mathematical expression.
Let’s take limits of functions for example. Definition 9.3.6 seems to suggest that we should write “function signs” after the symbol “ $\lim_{x\rightarrow x_0;x\in E}$ “. (I tend to view function signs as a special kind of mathematical expressions. In particular, a function sign should be of the form $\Theta(x)$ , where $\Theta$ is a function).
Now suppose that $f:X\rightarrow\mathbb{R}$ and $g:X\rightarrow\mathbb{R}$ are functions which converges to $L$ and $M$ respectively at $x_0$ in $E$ , then we write $\lim_{x\rightarrow x_0;x\in E}f(x)=L$ and $\lim_{x\rightarrow x_0;x\in E}g(x)=M$ . And we can prove the fact that their sum $f+g:X\rightarrow\mathbb{R}$ converges to $L+M$ , for which we write $\lim_{x\rightarrow x_0;x\in E}(f+g)(x)=L+M$ . I’m totally fine with this.
What confuses me is that people usually don’t write $\lim_{x\rightarrow x_0;x\in E}(f+g)(x)=L+M$ , but they write $\lim_{x\rightarrow x_0;x\in E}f(x)+g(x)=L+M$ instead. But we should write “function signs” after “ $\lim_{x\rightarrow x_0;x\in E}$ “. $(f+g)(x)$ is a function sign since $f+g$ represents a function, while $f(x)+g(x)$ is a mathematical expression (pertaining to an element $x\in X$ ), not a function sign. Why is it legitimate to write $\lim_{x\rightarrow x_0;x\in E}f(x)+g(x)$ ?
Similaly, I’m confused about the question of whether it is legitimate to write symbols such as $lim_{x\rightarrow 1}x^2=1$ , $\int cf(x)dx$ , $\sum_{x\in X}|f(x)|$ ,etc? For example, I’m fine with $\int (cf)(x)dx$ ,but $\int cf(x)dx$ seems confusing to me.
I know that this may not much sense, but I would really appreciate it if you could give me a hint.
Best regards

11 December, 2022 at 11:13 am

Terence Tao

Strictly speaking, limits such as $\lim_{x \to x_0; x \in E} f(x)$ are not functions of the quantity $f(x)$ per se, but of the operation $x \mapsto f(x)$ (also written $(\lambda x . f(x))$ in lambda calculus) that takes $x$ as input and returns $f(x)$ (thus converting $x$ from a free variable to a bound variable). But expressions such as $\lim_{x \to x_0; x \in E} (x \mapsto f(x))$ are ungainly and so we adopt the convention that expressions $f(x)$ that involve a free variable $x$ are identified with the operation $x \mapsto f(x)$ when placed inside an operation such as limit, sum, derivative, or integral that is treating that variable as bound instead of free.

This distinction between an expression with a free variable and the operation formed by abstracting that free variable to be bound can help avoid novice mistakes in calculus such as “We know that $x^2=9$ when $x=3$ , so after differentiating in $x$ we conclude that $2x=0$ when $x=3$ , which is absurd”. Here the point is that while $x^2$ and $9$ are equal when $x=3$ , the operations $x \mapsto x^2$ and $x \mapsto 9$ are not equal over any larger domain of $x$ (and in particular on any neighborhood of $3$ , which is what is needed in order for the derivative operation to be well defined.)

17 December, 2022 at 1:34 pm

OKON SAMUEL (@okonsamuel50)

Hello Professor Tao,
I’m currently reading the third edition of the text. And I must say it pretty easy to understand (so far though). I feel like I’m discovering things myself. lol
I noticed something off, which I thought might be an omission.
In proposition 3.6.14 (f)(Cardinal Arithmetic), shouldn’t there be an additional assumption that at least one of $X$ or $Y$ is non-empty. Because if both $X$ and $Y$ are empty, then, $\#(Y) = 0$, $\#(X) = 0$, and $\#(Y^X) = 1$ (since functions are uniquely defined by domain, range and graph). This results in the statement that $1 = 0^0$ which I taught was meant to be undefined.
I don’t know if I messed up.

18 December, 2022 at 5:49 pm

Terence Tao

$0^0$ is generally defined to equal $1$ when the exponent is interpreted as a natural number, but not when the exponent is interpreted as a real number. See https://en.wikipedia.org/wiki/Zero_to_the_power_of_zero

25 December, 2022 at 8:55 am

Windrunner

In the appendix on ill-formed statements you write: “Thus well-formed statements can be either true or false; ill-formed statements are considered to be neither true nor false.”

It seems that there is a problem here. For instance, the following is a true proposition: a/b>0 if and only if a,b>0 or a,b1 \Longleftrightarrow 0<x<1 when x=0.

Professor, I am really concerned about such undefined statements, if considered neither true nor false. I'm afraid they would cause chaos in our traditional two-valued logic.

Finally I hope it isn't a dumb question. Thanks.

25 December, 2022 at 10:13 am

Windrunner

Oh, I’m shocked. This time no LaTeX is used but the errors are exactly the same. Maybe it’s a WordPress bug. Now I have no other way but to upload a picture. Please see: https://postimg.cc/7Jkf8QWB Thanks.

25 December, 2022 at 6:11 pm

Matthew

I try to restate your proposition in a more precise way: “for all a∈R and for all b∈R\{0}: a/b＞0⇔(a,b＞0) or (a,b＜0)”. Then no confusion is caused.
When we say things like “P(x) is true for all x of type T”, we have to make sure that given any x of type T , P(x) is well-defined (i.e., P(x) is either a true statement or a false statement), otherwise the sentence “P(x) is true for all x of type T” itself would be ill-formed.
So “statements” such as “for all a∈R and for all b∈R， a/b＞0⇔(a,b＞0)or(a,b＜0)” would be considered ill-formed (I assume that this sentence is exactly what you’re trying to say). It doesn’t satisfy the principle mentioned in the second paragraph.

26 December, 2022 at 4:48 am

Windrunner

Thanks for the comment. I totally understand what you mean, but I’m not sure that is the right explanation. Let’s wait and see Professor Tao’s answer.

26 December, 2022 at 6:25 am

Matthew

Here’s an example that may be helpful to you.
In a previous comment, Professor Tao has mentioned some convention we adopt when dealing with propositions that don’t seem to have a natural truth value to be assigned to. For instance, let’s consider the conditional statement “if P, then Q”. What if P is false while Q is ill-formed/undefined? Actually, the sentence is considered to be true in this case.
I talk about this because there is an application of it which is so similar to your question: is “if $x=y$ and $z\neq 0$ , then $x/z=y/z$ ” true or false when $z=0$ ? Confusion is caused here because the hypothesis ( $x=y$ and $z\neq 0$ ) is false and the conclusion ( $x/z=y/z$ ) is ill-formed. However, by adopting the above convention, one can confidently claim that this is true for all choices of parameters.
Note that if one adheres strictly to the formal rules of first-order logic, one should claim that “if $x=y$ and $z\neq 0$ , then $x/z=y/z$ ” is true for all choices of x, y, z. The emphasis is on “for all choices of x, y, z”. In other words, the sentence “ $\forall x,y,z\in\mathbb{R}$ :if $x=y$ and $z\neq 0$ , then $x/z=y/z$ ” is a ture statement, while the sentence “if $x=y$ and $z\neq 0$ , then $x/z=y/z$ ” itself has no truth value, strictly speaking. The former is a sentence without free variables, the latter is a sentence with three free variables. However, it makes sense to claim that “if $x=y$ and $z\neq 0$ , then $x/z=y/z$ ” is true in practice (informally, I think). I think this is what Walter Rudin calls a “tacit convention” in his book.
Similaly, the sentence that confuses you (“a/b＞0⇔(a,b＞0)or(a,b＜0)”) is a sentence that contains two free variables. Stricly speaking, we need to add some quantifiers to get rid of the free variables so that a truth value can be obtained. For instance, we may claim that a/b＞0⇔(a,b＞0)or(a,b＜0) for all a∈R and for all b∈R\{0}.

26 December, 2022 at 12:30 pm

Windrunner

It so happens that I’ve read that comment too. It tells us “if X, then Y” is considered to be true when X is false, even if Y is undefined. However in the statement “a/b＞0 ⇔ (a,b＞0) or (a,b＜0)”, it only solves half of the problem.

That is to say, we can be sure that the following statement is true: for all a∈R and all b∈R (a,b＞0) or (a,b＜0) ⇒ a/b＞0.

Here the question is, for the other direction, can we say that for all a∈R and for all b∈R a/b＞0 ⇒ (a,b＞0) or (a,b＜0), or just for all b∈R\{0} the implication is true?

I see that you prefer the latter.

3 January, 2023 at 7:23 am

Matthew

Dear Professor Tao,
In Axiom 3.6 (Replacemet), we have a property P(x,y) pertaining to an object x in A and an object y. I think the symbolism P(x,y) is intended to indicate that x and y are both free variables in the formula P(x,y); in other words, both x and y should occur in P(x,y) at least once without being introduced by one of the phrases “for some x (y)” or “for all x (y)”.( Same for the property P(x) in the axiom of specification and the property P(x,y) in the definition of functions).
But in example 3.1.32, P(x,y) is the statement “y=1”, in which the letter x doesn’t occur. But I think that x should occur at least once (as a free variable). Is their any misunderstanding?
Best regards.

7 January, 2023 at 8:00 am

Terence Tao

It is permitted for a free variable such as $x$ to appear zero times in a property such as $P(x,y)$ . (Sometimes you will see this indicated through notations such as “Let $P(x,y) = P(y)$ be the property that $y=1$ “, to stress that $x$ is being considered as a free variable even though it appears zero times in the definition of the property.)

11 February, 2023 at 12:56 am

Anonymous

Professor, there might be an error in the errata to the corrected third edition (page 148, general definition of subsequence). The correct thing to say is $\left(b_n\right)_{n=m'}^\infty$ is a subsequence of $\left(a_n\right)_{n=m}^\infty$ iff there exists a strictly increasing function $f:\{n\in\mathbb{Z}:n\geq m'\}\rightarrow\{n\in\mathbb{Z}:n\geq m\}$ such that $b_n=a_{f(n)}$ for all $n\geq m'$ .
I don’t know if this has been corrected in the fourth edition. I hope this comment helps.

[Thanks, this has been added to the errata – T.]

14 February, 2023 at 1:38 am

Jonas

Possible erratum: Exercise 4.4.3 asks the reader whether the axiom of choice is necessary for a specific proof, but at this point the axiom of choice has not been introduced yet (only mentioned in passing).

[Fair enough; this has been added to the errata – T.]

18 February, 2023 at 3:22 pm

Radical_Drift

On page 234, it seems that there is some discussion on “asymptotic discontinuities.” Is this meant to be more of an informal discussion? In the formal definition, discontinuities presuppose that the point is in the domain of the function.

Basically, one way to phrase this is that, it’s a little weird to talk about 1/x being discontinuous at 0 if it’s not defined there (given the formal definition stated). On the other hand, “asymptotic discontinuities” (or infinite discontinuities) are pretty natural things, and the definition is not hard to state.

I guess it would be nice if you could elaborate on that discussion (I don’t have the book in front of me, I just remember the page number).

20 February, 2023 at 1:46 am

Robert Dicks

Whoops, I guess my comment applies to the third edition.

24 February, 2023 at 7:50 am

дэн (@dan_abramov)

Errata for Lemma 5.6.6 (e) on page 122 currently says:

>One can also replace x<1 by 0 < x 0 for (e).

At the beginning of Lemma 5.6.6, x is defined to be >= 0. If we plug x = 0 into (e), it would say 0^1/k is an increasing function of k. But that’s not true: 2 > 1, but 0^1/2 = 0^1/1. So I think x *needs* to be > 0.

Thanks!

24 February, 2023 at 7:53 am

дэн (@dan_abramov)

Oops, I think might comment got a bit borked, apologies. What I meant to say is that I think “One can also replace […] for clarity” should be “One should also replace […]” because x being greater than 0 seems like a required condition that’s not posited anywhere else.

26 February, 2023 at 6:40 am

Yingyuan Li

Dear Professor Tao,
I think I might have some possible suggestions on the content of Theorem 8.8.2 (Fubini’s theorem for infinite sums) in Analysis I 4th edition, page 165.
In Theorem 8.8.2 in Analysis I, it states that we can switch the order of infinite sums provided that the entire sum is absolutely convergent. However in usual practice, it is not so convenient to find a bijection between $\mathbb{N}\times\mathbb{N}\rightarrow\mathbb{N}$ , neither to prove the absolute convergence of $\sum_{(n,m)\in\mathbb{N}\times\mathbb{N}}{f(n,m)}$ . What we commonly do is to prove ” $\sum_{m=0}^{\infty}{f(n,m)}$ is absolutely convergent for all $n\in\mathbb{N}$ , and $\sum_{n=0}^{\infty}\sum_{m=0}^{\infty}{|f(n,m)|}$ is convergent “, which can be defined as absolute convergence of the double sum $\sum_{n=0}^{\infty}\sum_{m=0}^{\infty}{f(n,m)}$ .
On the other hand, we can prove that $\sum_{(n,m)\in\mathbb{N}\times\mathbb{N}}{f(n,m)}$ is absolutely convergent iff $\sum_{n=0}^{\infty}\sum_{m=0}^{\infty}{f(n,m)}$ is absolutely convergent.
Hence, I might suggest that we could add a bit these contents to the theorem 8.8.2 in Analysis I since in the following chapters we basically use this theorem in the form of the “double sum” rather than the “whole sum”.
And I have left a reply about some possible errata in Analysis II related to this Fubini’s theorem.
Thanks!

26 February, 2023 at 4:59 pm

Anonymous

When you say “ $\sum_{n=0}^{\infty}\sum_{m=0}^{\infty}|f(n,m)|$ is convenient”, it means $\sum_{(n,m)\in\mathbb{N}\times\mathbb{N}}f(n,m)$ is absolutely convergent.

26 February, 2023 at 9:56 pm

Yingyuan Li

These two kinds of convergence are logically equivalent, though a little bit proof should still be needed because the definitions of these two are indeed different. So I thought a bit more introduction about these might let the textbook be more logically integrated and beginner-friendly.

28 February, 2023 at 8:24 am

Yingyuan Li

Sorry again for typo!
It is 8.2.2 rather than 8.8.2.
Thanks!

27 February, 2023 at 4:37 am

Yingyuan Li

Dear Professor Tao,

In Definition 1.4.1 in Analysis II 4th edition page 15, would it possibly be better if “increasing” be replaced by “strictly increasing”, so that it would also be consistent with of definition 6.6.1 of subsequences in Analysis I?

All the best!

[Erratum added, thanks – T.]

27 February, 2023 at 5:40 am

Yingyuan Li

Dear Professor Tao,

In the errata for Analysis I 3rd ed, an erratum states that “Page 261: In 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.”, which subsuqently appears in the 4th ed, page 226.
But I have found that the addition of that “ $x_0,y_0$ are limit points of $X,Y$ respectively” could be omitted，since it could be implied by the other hypotheses that $f$ is invertible and differentiable at $x_0$ “.
$f$ is differentiable at $x_0$ . By the definition of differentiability, $x_0$ must be a limit point of $X$ . And differentiability also implies that $f$ is continous at $x_0$ . For any $\varepsilon>0$ , by continuity of $f$ at $x_0$ , there exists a $\delta>0$ , such that $\left|f(x)-y_0\right|\le\varepsilon$ for all $x\in$ and $\left|x-x_0\right|\le\delta$ . And by $x_0$ is a limit point of $X$ , for this $\delta$ , there exist an $x'\in X-\left\{x_0\right\}$ and $\left|x'-x_0\right|\le\delta$ . By $f$ is invertible, $f(x\prime)\neq y_0$ . Thus there exist a $y=f(x')$ such that $y\in Y-\left\{y_0\right\}$ and $\left|y-y_0\right|\le\varepsilon$ . Because $\varepsilon$ is arbitrary, it means that $y_0$ is a limit point of $Y$ .
Thus, for the sake of minimal set of hypotheses, this added hypothesis could be omitted.

Thanks!

[I prefer to leave these hypotheses in for clarity. -T]

28 February, 2023 at 7:27 am

Yingyuan Li

These two errata should be in Analysis II:

Page 18: In Definition 1.4.1, “increasing” should be “strictly increasing”.
Page 89: In the proof of Lemma 4.7.3, “ $\sin$ increasing” should be “$ latex\sin $ is strictly increasing”.

Thanks!

[Errata moved – T.]

7 March, 2023 at 8:43 pm

Gaurav Chandan

Dear Professor Tao,

On page 68 in the fourth edition, in Exercise 3.5.12, shouldn’t the domain of the function $a$ be $\mathbf{N}$ and not $X$ ?

[Correction added, thanks – T.]

9 March, 2023 at 12:25 am

Anonymous

Hi Dr Tao,

Could you give a hint as to how to solve part c of exercise 6.4.12? The exercise asks to prove that if $$c$$ is a limit point of a sequence $$(a_n)_{n=m}^\infty$$, then $$L^-\le c\le L^+$$. I have spent way too long(around 2 weeks) and can’t seem to figure out a way to prove it.

Thanks

[Try a proof by contradiction – T.]

11 March, 2023 at 6:58 am

Yingyuan Li

Dear Professor Tao:
Here might be some minor errata or suggestions.

An erratum to an erratum in Analysis 4th ed:
“Page ???: At the end of Section 8.2, add the following exercise (Exercise 8.2.7): “Let $f: {\bf N} \times {\bf N} \to {\bf R}$ be a function. Show that $\sum_{(n,m) \in {\bf Z}^2} f(n,m)$ is absolutely convergent if and only if $\sum_{n=1}^\infty \sum_{m=1}^\infty |f(n,m)|$ is convergent.””
Here ??? should be 170 (Springer 4th ed), $\sum_{(n,m) \in {\bf Z}^2} f(n,m)$ should be $\sum_{(n,m) \in {\bf N}^2} f(n,m)$ , and $\sum_{n=1}^\infty \sum_{m=1}^\infty |f(n,m)|$ should be $\sum_{n=0}^\infty \sum_{m=0}^\infty |f(n,m)|$ .

In page 169 (Analysis I, Springer 4th ed), in Theorem 8.2.8, it says “Let $\sum_{n=0}^\infty a_n$ be a series which is conditionally convergent.” Would it possible be better if we emphasize that “Let $\sum_{n=0}^\infty a_n$ be a series of real numbers which …”?

Best regards!
Y.

[Corrected, thanks – T.]

11 March, 2023 at 3:23 pm

Anonymous

Hi Dr Tao,

I have tried proving part c of exercise 6.1.12 by contradiction and haven’t been able to. Would it be possible for you to give more hints?

Thanks

12 March, 2023 at 4:35 pm

Anonymous

I actually meant part d and not part c. I have already done part c.

27 March, 2023 at 5:41 am

Yingyuan Li

Dear Professor Tao:

Here might be a minor erratum.

In page 261 (Analysis I, Springer 4th ed), Remark 11.10.4, $f\frac{d\alpha}{ }dxdx$ should be $f\frac{d\alpha}{dx}dx$ .

Best regards!
Y.

[Strange – in my version it already appears as the latter -T.]

27 March, 2023 at 12:20 pm

Yingyuan Li

Dear Professor Tao:

Here might be some possible errata or suggestions.

In page 252-253 (Analysis I, Springer 4th ed), Definition 11.8.1:
“… let X be a interval that is closed …”, here “a” should be “an”;
In (ii), “… when X is the right-endpoint (resp. left-endpoint) of X).” here the last right parenthesis “)” should be deleted, and the first “X” should be “a”, so the whole sentence should be “… when $a$ is the right-endpoint (resp. left-endpoint) of $X$ .”
In (iii), $\alpha[I]:=\lim_{x \rightarrow b- : x \in X}{\alpha(x)} - \lim_{x \rightarrow b+ : x \in X}{\alpha(x)}$ should be $\alpha[I]:=\lim_{x \rightarrow b- : x \in X}{\alpha(x)} - \lim_{x \rightarrow a+ : x \in X}{\alpha(x)}$ with “b+” replaced by “a+”.
In addition, I have found that the left and right limits symbols used in this definition are actually different from those in chapter 9.5 (e.g “:” would be “;” instead, and more distinctly, $\lim_{x \rightarrow b- : x \in X}{\alpha(x)}$ would be $\lim_{x \rightarrow b ; x \in X \cap (-\infty , b)}{\alpha(x)}$ instead. So may I make a possible suggestion that we use such symbols as $\alpha (b-)$ and $\alpha (b+)$ for the left and right limits instead for consistency with chapter 9.5 and also for easy-reading?

Best wishes!
Yingyuan Li

[Errata added – T.]

27 March, 2023 at 1:27 pm

Jonas Esser

Dear Prof. Tao,

I have been unable to prove that $(x^q)^r = x^(qr)$ when $x, q, r$ are real numbers with $x$ positive. This is part of Proposition 6.7.3; I was able to prove all the other identities.
I also looked in other textbooks (Rudin), but this was not helpful, since Rudin defines real exponentiation in terms of power series.
Do you (or anyone else reading this) have a hint or another source regarding this particular proof? Thank you.

Best wishes,
Jonas

27 March, 2023 at 1:30 pm

Jonas Esser

Sorry, I messed up the LaTeX there – the identity I mean is $(x^q)^r = x^{qr}$ .

28 March, 2023 at 4:55 am

Jonas Esser

To be a bit more precise, I managed to narrow things down to the conjecture that $\lim\limits_{n \to \infty} x^{q_n r_n} = \lim\limits_{k \to \infty} \lim\limits_{n \to \infty} x^{q_n r_k}$ , given that $x$ is a positive real number and $(q_n)_{n=1}^{\infty}$ , $(r_n)_{n=1}^{\infty}$ are convergent sequences of rational numbers. But I don’t know how to prove it.

28 March, 2023 at 7:37 pm

Yingyuan Li

I have the same feeling that this proof is complicated and one will easily fall into mistakes.
I have taken notes about this proof here: Notes: Basic Properties of Real Exponentiation –
https://zhuanlan.zhihu.com/p/617784818
I hope it can be normally displayed since it is a Chinese website and thus will help.
I not sure if it is proper to share my notes here in professor’s blog. If not, please deleted this reply! Thanks.

28 March, 2023 at 11:43 pm

Jonas Esser

Thank you very much for your reply! This was helpful indeed. Your “real version of lemma 6.7.1” was the missing ingredient in my proof.
I was able to prove the chain of identities

$(x^q)^r = (\lim\limits_{n \to \infty} x^{q_n})^r = \lim\limits_{k \to \infty} (\lim\limits_{n \to \infty} x^{q_n})^{r_k} \\ = \lim\limits_{k \to \infty} (\lim\limits_{n \to \infty} (x^{q_n})^{r_k}) = \lim\limits_{k \to \infty} (\lim\limits_{n \to \infty} x^{q_n r_k}) = \lim\limits_{k \to \infty} x^{q r_k}$ .

The only missing piece is showing that $\lim\limits_{k \to \infty} x^{q r_k} = x^{qr}$ . But this is now easy using your lemma.
My approach is a bit different from yours, but both work as far as I can see.

27 March, 2023 at 5:15 pm

Benjamin Bachrach

Hi Professor,

I am reading the fourth edition. Here are 2 minor corrections I have noticed:

1) On page 5, “theirbooks” should say “their books”
2) On page 10, when describing that a limit diverges, an exponent is listed as “-2” instead of “-3”.

Thank you for your work to help folks like me learn mathematics.

Benjamin

[Corrections added, thanks – T.]

27 March, 2023 at 5:16 pm

Benjamin Bachrach

I should add this is for Analysis I

27 March, 2023 at 5:25 pm

Benjamin Bachrach

Further context: this is for the Hindustan Book agency version, Examples 1.2.5 and 1.2.12.

29 March, 2023 at 6:41 am

Yingyuan Li

Dear professor Tao,

Here might be some possible suggestion or minor correction.

In page 252-253 (Analysis I, Springer 4th ed), in the definition of α-length, we use the left and right limits. So can we add a bit content after the definition or an exercise at the end of this chapter to show that these limit do exist for monotonic functions and thus this definition is well-defined? Possibly like this:
Excercise 11.8.??:
Let $X$ be a closed subset of $\mathbf{R}$ , and $\alpha : X \rightarrow \mathbf{R}$ be an monotonic function. Let $x_0$ be a real number and also an adherent point of $X \cap (-\infty, x_0)$ (resp. $X \cap (x_0,+\infty)$ ). Show that the left limit (resp. right limit) of $\alpha$ at $x_0$ exists. And explain that the α-length is well-defined.

In page 253 (Analysis I, Springer 4th ed), in Lemma 11.8.4, “… is closed and which contains …” should be “… which is closed and contains …”.

Thanks and best regards!
Yingyuan

[Errata added – T.]

30 March, 2023 at 6:17 am

Yingyuan Li

Some more possible minor corrections:

In page 253 (Analysis I, Springer 4th ed),Definition 11.8.1 (iv)”α((a, b)) + α({b}), α({a}) + α((a, b)), or α({a}) + α((a, b)) + α({b})” should possibly be “α[(a, b)] + α[{b}], α[{a}] + α[(a, b)], or α[{a}] + α[(a, b)] + α[{b}]” .

Best regards!
Yingyuan

[Correction added, -T.]

2 April, 2023 at 7:04 am

Yingyuan Li

Dear Professor Tao,
Thank you again for your great work Analysis I and II that lead me into the mathematical world.

Recently I have been re-reading chapter 11 of Analysis I (The Riemann Integral) and thinking for quite a long time about the difference between your “new” definitions of RS integral and the traditional one (e.g. like those in Rudin’s Principles of Mathematical Analysis).

I seem to realize that the fundamental difference lies in the definition of “partition”. In the traditional one, it has overlapping of all sub-intervals at their endpoints. However, in your new definition, it just eliminates this overlapping. A result of this change is the new α-length, whose new definition is given a final version in the fourth edition of the textbook in order to match the new definition of partition.

Today I am a little bit exciting because I think I have figured out a proof showing that your new RS integrability is an extension of the traditional one rather than an equivalence, i.e. the traditional RS integrability can imply the new RS integrability, but the reverse does not hold . (The proof is too long to post here though.)

And I still have doults in a point. In the textbook it says “if f and α are both discontinuous at the same point, then $\int_{I}f \mathrm{d} \alpha$ is unlikely to be defined”. But it seem that this problem only occurs in the traditional RS. In Rudin’s textbook, it gives an exercise (ex 6.3: β(x)=0 when x0; and f is a bounded function defined on [-1,1]. Then f is RS integrable with respect to β iff f is continuous at 0. ), which shows that discontinuity at the same point will lead to non-integrability. However, with your new definition, it seems that it becomes integrable since we can have a partitiona P = {[-1,0) , {0} , (0,1]}. The “single-point” subinterval {0} just eliminates the problem that upper and lower RS-sums do not coincide. For example, if we define f(x)=1 when x∈[-1,0), f(x)=2 when x=0, f(x)=3 when x∈(0,1], then $\int_{I}f \mathrm{d} \beta = 2$ based on β[{0}]=β(0+) – β(0-) =1 and the fact that f is exactly the piecewise constant function both majorizes and minorizes itself. So discontinuity at the same point may not be a problem any more with the new definition.

I am not sure whether the above are right. And I am really looking forward to your comments.

Thank you very much and best regards！
Yingyuan Li

2 April, 2023 at 7:38 am

Yingyuan Li

I guess that the “less” and “greater” symbols made the errors.

The Rudin’s exercise is like this:

(ex 6.3: β(x)=0 when x∈(-∞,0), β(x)=1/2 when x=0, β(x)=1 when x∈(0,+∞); and f is a bounded function defined on [-1,1]. Then f is RS integrable with respect to β iff f is continuous at 0. )

25 April, 2023 at 5:08 am

Jonas Esser

About Proposition 8.1.10: Let $X$ be a countable set, and let $Y$ be a countable set. Then $X \cup Y$ is a countable set.

I think the assumptions could be weakened, at the expense of a longer proof: if $Y$ is at most countable (i.e. possibly finite), the result still holds.

1 May, 2023 at 10:28 pm

Felix B.

Dear Professor Tao,

I think on page 38 (Section 3.2 Russell’s Paradox (Optional)) of the fourth edition it should read “…, then by the definition of $P$ $P(\Omega)$ would be FALSE, and hence by the definition of $\Omega$ we have $\Omega\in\Omega$.”

Personally, I think the argument made becomes clearer when one writes “…, then by the definition of $P$ $P(\Omega)$ would be false, and, since $\Omega$ is a set, we have $\Omega\in\Omega$.”

On a final note, am I correct when I say that the Zermelo-Fraenkel axioms you are stating are actually the ones allowing for atoms (individuals, urelements)?

Thank you and best regards,
Felix

6 May, 2023 at 12:45 am

Anonymous

Dear Professor Tao, I have a question about exercise 11.10.4. If phi is monotone decreasing, it seems to conflict with the definition of alpha length. How can this be resolved?

[To obtain an appropriate analogue, one will have to modify the definition of alpha-length to handle monotone decreasing functions. -T]

11 May, 2023 at 5:29 am

Anonymous

Dear Professor Tao,

In Analysis I Chapter 4, ‘supersede’ is misspelled as ‘supercede’ in pages 77, 79, 82, 84, and 90.

Thanks!

12 May, 2023 at 8:02 am

Carlos Martínez

Dear Professor Tao,

There may be a possible erratum on page $51$ of the fourth edition, in the second part of exercise $3.4.6$ . It is virtually identical to exercise $3.5.11$ , but the necessary machinery to construct ordered pairs (and therefore functions) using sets has not been introduced yet. Furthermore, in that very same exercise, a ‘from’ should be added between ‘deduced’ and ‘the’.

Best regards,
Carlos

[Fair enough; I’ve added an erratum to delete this part of the exercise as being redundant. -T]

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