Analysis I

Last updated: Dec 8, 2025

Analysis, Volume I
Terence Tao
Hindustan Book Agency, January 2006. Fourth edition, 2022. (Also Springer, Fourth edition, 2022.)
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 official solution guides for this text. However, there is a Lean companion.

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 ???: In Exercise 5.6.5, replace “positive rational $q$ with $q > 1$ ” with “rational $q" with$ latex q>0$”, and at the end of the exercise, ask what happens if $q<0$ or $q=0$ (rather than $q<1$ ).
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 ???: In the hint for Exercise 6.6.5, “each natural numbers” should be “each natural number”.
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 $X$ “. In Exercise 8.5.19, the := should just be an =.
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 “ $f^{-1}$ 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, impose the hypothesis that X be non-empty. (The reason for these changes is to be consistent with the notion of connectedness used in Analysis II and in other standard texts. -T.)
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 ???: At the start of the proof of Proposition 11.6.1, add “We may assume that $a \leq b$ , since the claim is vacuously true otherwise.”.
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”. Colons in maps should be spaced using the \colon LaTeX macro.
Page x: “Chapter 5 (on Fourier Series)” should be “Chapter 5 of Analysis II (on Fourier Series)”.
Page 5: “theirbooks” should be “their books”.
Page 9: In Example 1.2.12, final paragraph, $4 x^{-2}$ should be $4 x^{-3}$ . In footnote 1, “also continuous and differentiable” should be “also continuous and partially differentiable in the x,y directions”
Page 12: “carry of digits” should be “carry digits”.
Page 14: the computing language C should not be italicised.
Page 15: The hyperlinks in Footnote 4 should point to Appendix A. (This only affects certain electronic versions of the text.)
Page 16: The semicolon before $5 \times 3$ should be a colon.
Page 16: In Remark 2.1.10, “axi” should be “axiom”. (This only affects some versions of the text.)
Page 17: In the parenthetical, “ $n$ “is not a half-integer”” should be “” $n$ is not a half-integer””, i.e., the $n$ should be inside the quotes.
Page ???: In the reference to Lemma 8.5.15, “transfinite induction” should be “Zorn’s lemma”.
Page 19: In Remark 2.1.5, the first “For instance” may be deleted.
Page ???: In Remark 2.1.14, “as to argue about” should be “to argue about”.
Page ???: In Proposition 2.1.16, “assign a unique natural number” can be rephrased as “uniquely assign a natural number” to reduce ambiguity.
Page 29: In Axiom 3.3, both versions of the emptyset should be $\emptyset$ . (This only affects some versions of the text.)
Page 30: In Example 3.1.10: “lie on” should be “lie in”.
Page 31: In the last part of Definiton 3.1.14, “if” should be “iff”.
Page 40: In Remark 3.1.26, “elementsof” should be “elements of”.
Page 42: In Definition 3.3.8, the first “if” should be “if and only if”.
Page 46: In the parenthetical to Exercise 3.3.1, replace “are immediate from … in question, but the point” with “would be immediate from … in question, but as discussed in Remark 3.3.8, the axioms of equality for functions require justification. The point…”
Page ???: In Exercise 3.4.3, “next section” should be “next chapter”.
Page 49: 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”.” Also, “can be deduced the preceding axioms” should be “can be deduced from the preceding axioms”.
Page 51: In Remark 3.4.13, “Ernest” should be “Ernst”.
Page 55: The spelling “rôles” is no longer in common use and can be replaced with “roles”.
Page 56: In Exercise 3.5.6, “the $A,B,C,D$ ” should be “the sets $A,B,C,D$ “. In Exercise 3.5.12, $a$ should be a function from ${\bf N}$ to $X$ , and $a_N(n++) = f(n,a(n))$ should be $a_N(n++) = f(n,a_N(n))$ . In Exercise 3.5.13, the function $f$ can be renamed $a$ for notational consistency with Exercise 3.5.12.
Page 62: In the hint for Exercise 3.6.12(i), the codomain for $\phi$ should involve the range $1 \leq i \leq n++$ rather than $1 \leq i \leq n$ , and the set after “from the set” should be $S_{n++}$ rather than $\{ i \in {\mathbf N}: 1 \leq i \leq n++ \}$. Also, two of the left parentheses in this exercise need to be closed up with matching right parentheses.
Page 64: In footnote 1, “two applications of the axiom of replacement” should be “the axiom of replacement”, and “role” should be “rôle”.
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 ???: Add an exercise to Section 4.1: “Suppose we define the integers set-theoretically as $a -\!- b := \{ (c,d) \in {\mathbf N}^2: a+d=b+c \}$ as suggested in Footnote 1. Show that this definition is consistent with Definition 4.1.1 in the sense that, when one adopts this definition, one has $a -\!- b = c -\!- d$ if and only if $a+d=b+c$ .
Page 74: 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 79: Delete the last sentence of Exercise 4.4.3 (as the axiom of choice has not yet been formally introduced.)
Page 87: In Remark 5.2.4 “Oepsilon-close” should be “epsilon-close”.
Page 88: In the sixth line from the bottom of the proof of Proposition 5.2.8, delete the first “yet”.
Page 89: In the statement of Lemma 5.3.6, delete the space before the close parenthesis.
Page 94: 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 ???: In Exercise 5.3.3, “equivalent” should be “are equivalent”.
Page 97: After Proposition 5.4.9, “Section 1.2” should be “Section 1.7 of Analysis II”.
Page 98: The paragraph after Remark 5.4.11 may be deleted, since it is essentially replicated near Definition 6.1.1.
Page ???: In Exercise 5.4.9(ii), add “If”.
Page ???: In the proof of Proposition 5.5.8, $M_1$ and $M_2$ are introduced twice; one of these redundant introductions may be deleted.
Page 103: In Remark 5.5.15 “the greatestlower bound” is missing a space.
Page ???: In Lemma 5.6.5, delete the requirement that $q > 1$ , and in the final part, ask instead what happens when $q$ is negative.
Page ???: After Definition 6.2.1, the proposed choices for $+\infty$ and $-\infty$ can be simplified slightly to $+\infty := {\mathbf R}$ and $-\infty := {\mathbf R} \cup \{+\infty\}$ if desired.
Page ???: In Definition 6.4.6, one needs to extend the notion of supremum and infimum to cover sequences that take values in the extended reals, not just the reals.
Page ???: In Example 6.4.9, “infimumof” is missing a space.
Page ???: In the proof of Corollary 6.5.1, “Lemma 5.6.6” should be “Lemma 5.6.9(d)”.
Page 128: After Definition 6.6.1, “for all $n \in {\bf N}$ ” should be “for all $n \geq m'$ “.
Page 131: In the proof of Lemma 6.7.1, remove absolute values from second display for consistency.
Page ???: In the displays in the proof of Proposition 7.1.8, $x$ should be $f(x)$ in several places.
Page ???: In Proposition 7.1.11(e), “ $f: X \cup Y \to {\mathbf R}$ is a function” should be “ $f: X \cup Y \to {\mathbf R}$ be a function”. In Exercise 7.1.4, one can refer to Exercise 3.6.12(ii) for the definition of the factorial function.
Pages 167-168: In Lemma 8.2.3 and Definition 8.2.4, $\infty$ should be $+\infty$ .
Page 136: In Lemma 7.1.4(a), the condition $n < p$ may be relaxed to $n \leq p$ and $m \leq n$ to $latex m \leq n+1$, and in parts (b)-(f), the condition $m \leq n$ may be deleted.
Page ???: In Proposition 7.2.13(c), “series are convergent” should be “series is convergent”.
Page ???: Replace Exercise 7.2.1 with “In view of Example 7.2.7, can you now resolve the difficulty in Example 1.2.2”?
Page 152: In Exercise 7.3.2, add the requirement $x \neq 1$ to the geometric series formula.
Page ???: In the proof of Corollary 7.3.7, Lemma 6.7.3 should be Proposition 6.7.3.
Page ???: Exercise 7.3.3 can be moved to Section 7.2.
Page 152: 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 153: At the end of Example 7.4.2, “Exercise 5.5.2” should be “Exercise 5.5.2 of Analysis II”.
Page 154: In Example 7.4.4, “see Example 4.5.7” should be “see Example 4.5.7 of Analysis II”.
Page ???: In the last sentence of Exercise 7.4.1, “f” is not in math mode.
Page ???: In the proof of Lemma 7.5.2 (third paragraph, third sentence), “without” should be “Without”. The parenthetical “(why? prove by contradiction)” can be replaced by “(see Exercise 5.4.7)”.
Page ???: In Definition 8.2.1, = should be :=.
Page 166: In the proof of Theorem 8.2.2 just before the second display, “convergent for each $m$ ” should be “convergent for each $n$ “.. In the second display, the first $\leq$ sign can be replaced with equality. Two lines later, “Takingsuprema of this” is missing a space. The parenthetical “(Why will this be enough? prove by contradiction)” can be replaced by “(see Exercise 5.4.7)”.
Page ???: In Proposition 8.2.6, permute part (a) to follow after parts (b) and (c), rather than before, since parts (b), (c) can be helpful to prove (a).
Page ???: At the end of the statement of Lemma 8.2.7, add “and in fact diverge”.
Page ???: In the proof of Theorem 8.2.8, in the paragraph after (II), delete “ $n_j := \min$ ” before $\{ n \in A_-: n \neq n_i \hbox{ for all } i < j \}$ .
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 ???: In the proof of Corollary 8.3.4, next to $n \in A,B$ , add a footnote “Here we adopt the common convention of using $n \in A,B$ as an abbreviation for “ $n \in A$ and $n \in B$ “, and similarly for $n \not \in A,B$ .”
Page 173: In Remark 8.3.5, “Exercise 7.2.6” should be “Exercise 7.2.6 of Analysis II“. In Exercise 8.3.2, “a injection” should be “an injection”. In Remark 8.3.6,, replace Exercise 3.6.7 with Exercise 3.8.4.
Page 174: In Example 8.4.2, replace “For any sets $I$ and $X$ ” with “For any set $X$ and non-empty set $I$ “.
Page 174: In the first paragraph of Section 8.4, “Section 7.3” should be “Section 7.3 of Analysis II“.
Page ???: In Definition 8.4.1, = should be :=.
Page ???: In Remark 8.4.6, “Definition 1.2.12” should be “Definition 1.2.12 of Analysis II”. In Exercise 8.4.2, “choice of sets” should be “choice of set”, and “trick by” should be “trick of”.
Page ???: In Exercise 8.5.7, “maximum” and “minimum” can be replaced by “maximal element” and “minimal element”. Then add “In view of this result, for totally ordered sets, one can refer to “the maximum” instead of “maximal element” (if it exists), and similarly for “minimum”.
Page 178: In Proposition 8.5.10, replace $<_X$ with $<$ .
Page 180: In the proof of Lemma 8.5.14, a right parenthesis is missing for $s(\{y \in Y_\infty\cup \{s(Y_\infty)\}: y < x \}$ in the paragraph starting with “We now claim that…”.
Page ???: In Exercise 8.5.10, the set $Y$ in the hint can be replaced with the simpler $\{ n \in X: P(n) \hbox{ false} \}$ .
Page 181: 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 182: In Exercise 8.5.20, “a subcollection $\Omega' \subset \Omega$ ” is missing a space.
Chapter 9 in general: many references to $\infty$ should be to $+\infty$ .
Page 186: In the final paragraph of the proof of Lemma 9.1.12, $(a-b)$ should be $(a,b)$ .
Page ???: In the hint for Exercise 9.1.4, “axiom of choice” should be “the axiom of choice”.
Page 187: In Remark 9.1.25, “Theorem 1.5.7” should be “Theorem 1.5.7 of Analysis II“, and “Section 1.5” should be “Section 1.5 of Analysis II”.
Page ???: In the final claim of Proposition 9.3.14, add “with the convention that $f(x)/g(x)$ is set to some arbitrary value when $g(x)=0$ (which can happen for $x \not \in E$ )”.
Page 194: In Remark 9.3.15, for completeness one can also include “ $\lim_{x \rightarrow x_0} c f(x) = c \lim_{x \rightarrow x_0} f(x)$ ”.
Page ???: In Exercise 9.4.1, “(a),(b)” needs a space.
Page 197: In Example 9.4.3, in the second limit, $x_0 \in x$ should be $x \to x_0$ .
Page 199: In the paragraph after Proposition 9.4.11, “see Exercise 4.5.10” should be “see Exercise 4.5.10 of Analysis II”. 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 ???: In Exercise 9.4.1, the hint should be updated to account for part (d).
Page ???: In Exercise 9.6.1(c), the period at the end should be a semicolon.
Page 205: In the proof of Proposition 9.6.7, “in Proposition 2.3.2” should be “… Proposition 2.3.2 of Analysis II”.
Page 206: At the start of Section 9.7, “a continuous function attains” should be “a continuous function on a closed interval attains”.
Page ???: In the hint for Exercise 9.8.5(c), the argument for $f_n$ should be changed from $x$ to $y$ to avoid collision.
Page 211: In the second paragraph of Section 9.9, the “island of stability” can be defined as a closed interval rather than an open one for consistency with the rest of the text. In the first paragraph of Section 9.9, “continuous at $(0,2)$ ” should be “continuous on $(0,2)$ “.
Page 212: The note that we no longer require the sequences to be Cauchy can be safely deleted.
Page 214: In the first display of the proof of Theorem 9.9.16, $n \in N$ should be $n \in {\bf N}$ .
Page 215: In the first paragraph of Section 9.10, “see Section 11.12” should be “see Section 2.5 of Analysis II”.
Page 216: In Example 9.10.4, $(0, \infty)$ should be $(0, +\infty)$ (two occurrences).
Page 220: In Theorem 10.1.13(h), enlarge the parentheses around $\frac{f}{g}$ .
Page ???: In the last sentence above Proposition 10.3.3, “if function” should be “if a function”.
Page 227: In Exercises 10.4.1-10.4.3, replace $\infty$ with $+\infty$ throughout.
Page 232: In Remark 11.1.2, “Section 2.4” should be “Section 2.4 of Analysis II“.
Page 233-234: In the proof of Theorem 11.1.13, $I-K$ can be $I \backslash K$ for consistency.
Page 235: In Exercise 11.1.3, $c \leq b$ should be $c<b$ (two occurrences), and “ $I_j$ is not of the form” should be “none of the $I_j$ are of the form”. In the hint for Exercise 11.1.2, add “if nonempty” after “is automatically connected”.
Page ???: After Definition 11.3.4, add that an alternate notation for $\int_{[a,b]}$ is $\int_a^b$ .
Page 240: 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.) In Remark 11.3.8, “Chapter 8” should be “Chapter 8 of Analysis II“.
Page 246-247: In the proof of Theorem 11.5.1, the strict inequalities involving epsilon and delta can be replaced with non-strict inequalities for consistency with the rest of the book.
Page 251: Delete Corollary 11.6.5, and replace Exercise 11.6.5 with “Use Proposition 11.6.4 to provide an alternate proof of Corollary 7.3.7 in the case $q \neq 1$ , using the fundamental theorem of calculus (Theorem 11.9.4). The $q=1$ case can also be treated by this method, but requires the theory of the natural logarithm function, which is postponed to Analysis II“. This exercise should also be moved to Section 11.9.
Page 252: In Remark 11.7.2, “Chapter 8” should be “Chapter 8 of Analysis II“.
Page 253: 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, “interval $X$ is closed and which contains $I$ ” should be “closed interval $X$ containing $I$ “. After mentioning the continuous case, add as a parenthetical comment that while in this section one will only need the monotone case of this lemma, the continuous case will be useful in the next section.
Page 254: After Example 11.8.7, $\alpha(I)$ should be $\alpha[I]$ (two occurrences).
Page 255: At the end of Section 11.8, the reference to the breakdown of Theorem 11.4.1(g) should be deleted.
Page 257: 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 ???: In the statement of Proposition 11.10.1, the notation $I$ can be safely removed.
Page ???: In the proof of Theorem 11.10.2, replace “why is this generalization true?” with “see Exercise 11.4.2.”.
Page ???: In Remark 11.10.4, $f \frac{d\alpha}{} dx dx$ should be $f \frac{d\alpha}{dx} dx$ . (This issue may already be corrected in some printings.)
Page 271: In Examples A.2.1, “no conclusion on $x$ or $x^2$ ” should be “no conclusion for $x^2$ “.
Page 283-284: 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 332.
Page ???: In Exercise B.1.1, $\epsilon_{i+1}=1$ should be $\epsilon_{i+1} := 1$ .
Page 290: In the paragraph immediately after proof of Theorem B.1.4, “Onceone has this” is missing a space.
Page 291: In Definition B.2.1, note that the + sign is often omitted from the decimal representation.
Page 291-292: In the proof of Theorem B.2.2, many of the indices of $a$ need to have their sign flipped for consistency, e.g., $a_i$ should be $a_{-i}$ .
Page 293: In Exercise B.2.3, “ $x$ is not at terminating decimal” should be “ $x$ is not a terminating decimal”. $n/10^{-m}$ can be $n/10^m$ , and $m$ can be set to be a natural number rather than an integer.
In the index, the entries for ++ and for Cauchy-Schwarz are duplicated in some issues.
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, Rona Alenpour, James Ameril, Paulo Argolo, Paul Ashurov, William Barnett, José Antonio Lara Benítez, Dingjun Bian, Philip Blagoveschensky, Tai-Danae Bradley, Brian, Eduardo Buscicchio, Maurav Chandan, Hanson Char Diego Cimadom, Matheus Silva Costa, Gonzales Castillo Cristhian, Ck, William Deng, Kevin Doran, Lorenzo Dragani, Jonas Esser, Tejomay Gadgil, Evangelos Georgiadis, Elie Goudout, Ti Gong, Cyao Gramm, Christian Gz., Ulrich Groh, Yaver Gulusoy, Guanyuming He, Noel Hinton, Deniz İmge, 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, Olli Pottonen, 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, sotpau, Tim Smith, Sundar, suinwethilo, Karim Taha, Yigithan Tamer, 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.

1,450 comments

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19 April, 2025 at 6:43 am

Guanyuming He

Dear Prof. Tao,

Another question about real analysis and intuitionistic logic: I read here about the development of intuitionistic logic by Brouwer, and I learned that he (re)developed classical analysis intuitionistically, by using “choice sequences”, “spreads” and other constructs.

According to the article, intuitionistic logic seems to have some conflicts with classical logic. Brouwer was able to prove things that are not classically valid, like every total function on $[0,1] \to \mathbf{R}$ is continuous. Also, he showed that LEM would lead to a contradiction: $\forall x \in \mathbf{R} (P x \lor \neg P x)$ is not true, because it is false (in his analysis) that every real number is either rational or irrational.

However, at the end of the article, it writes

if one keeps to the letter of the classical theory but in its interpretation substitutes intuitionistic notions for their classical counterparts, one arrives at a contradiction. So it is not a counterexample in the strict sense of the word, but rather a non-interpretability result. As intuitionistic logic is, formally speaking, part of classical logic, and intuitionistic arithmetic is part of classical arithmetic, the existence of strong counterexamples must depend on an essentially non-classical ingredient, and this is of course the choice sequences.

I think I understand that for the Brouwer’s constructs in that article, but I don’t know which case it is for proof assistants. They are based on intuitionistic logic, but they are also used to formalize classical results. Does it mean they keep to the “letter of the intuitionistic theory” but their interpretation “substitutes classical notions“?

Thank you in advance.
Yours truly,
Guanyuming He

24 April, 2025 at 8:07 am

Guanyuming He

Dear Prof. Tao,

I have got quite a few of questions after I am exposed to intuitionistic logic and then went to reread your construction of natural numbers with that view in mind. Here, I want to ask one question that I think is the most prominent among them.

For your proof of 6 not equal to 2, I believe I could generalize it for any two natural numbers $m$ and $n := m+a'{++}$ for some $a'$ . That is, I want to give a proof of $\forall m,a' : \mathbf{N}, m \ne m+a'{++} =: n$ .

Step 0. Assume $m = n$ . We try to derive falsehood.
Step 1. Try to unwrap $m = m'{++}$ and $n = n'{++}$ . If it is not possible, then by the fact that natural numbers can only come from 0 and ${++}$ (using Axiom 2.5), we get $m = 0$ , then go to Step 3. Otherwise, go to Step 2.
Step 2. Use Axiom 2.4 to conclude $m' = n'$ . Then, use them to repeat Step 1.
Step 3. Now we arrive at $n'{++} = 0$ for some $n'$ , a contradiction to Axiom 2.3.

I want to ask, intuitionistically:
How could we be sure, for a general $m$ , that we can eventually arrive at Step 3? By definition, $m$ is 0 being applied ${++}$ m times. Is this enough to be sure of reaching Step 3, for a general $m$ , not a specific one, like 6 or 2?

One subtlety that makes me struggle to accept this is: If we use the intuitionistic philosophy that something is only true after we have experienced its truth, then, for some $m$ , even if we know it is 0 wrapped with ${++}$ for $m$ times, it does not automatically make we experience the truth that, after we unwrap it $m$ times, we reach 0, until we perform them. Moreover, to say that, because $n = m+a'{++}$ , we will always unwrap m first to 0 (an implicit assumption in Step 1), we also need to experience its truth by unwrapping. For a specific number, we can just do it and experience it. But to say that is true for all $m : \mathbf{N}$ , it is not so. I thought of using induction, but I feel a bit strange about it, because I don’t know if induction can help us experience the unwrapping for all natural numbers.

I see this is reflected in many such proofs, that’s why I feel this question is important. For another example, now suppose we accept that proof, and want to use that proposition as a lemma to prove the LEM of ordering of natural numbers (using your def): $\forall m,n : \mathbf{N}, (m \le n \lor \neg (m \le n))$ .

Step 0. We check if any of them is 0. If $m = 0$ , then we got $0+n = n$ and by definition $m \le n$ . If $n = 0 \land m \ne 0$ , then by the previous lemma, we conclude there’s no $a$ such that $m+a = n$ , that is, $m \le n$ is false. If neither is 0, the go to Step 1.
Step 1. If they are both not 0, then, we can say $m=m'{++} \land n=n'{++}$ . We then put $m', n'$ into Step 0. Here, we also need to use the principle talked about in my question to say that after some finite $a$ steps, one of them will become 0, and proceed to conclude the truth or falsehood of $m \le n$ .

Thank you in advance,
Yours truly,
Guanyuming He

25 April, 2025 at 6:45 am

Guanyuming He

Dear Prof. Tao,

I feel that what I said in my last question was not on the point, and the main point should not have to do with intuitionism. I have now came up with the correct way to express it:

The proof I gave in my last question is variable in length. How many times Step 1 is executed in the first depends on the value of $m$ . From what I learned (non-systematically), a formal definition of proof is a finite sentence that results from a formal language like the natural deduction calculus. The steps in my proof follow from the calculus, and what makes me wonder is its length that varies, depending on a parameter $m$ .

Do we count these variable length sentences as proofs, so long as the final result is finite?

I think that is the way how I wanted to express my previous question.

Thank you again,
Best,
Guanyuming He

25 April, 2025 at 8:54 am

Terence Tao

For these sorts of questions it is important to distinguish between the “internal” language of the theory one is studying (e.g., the theory of the natural numbers or of sets), and the “external” language of the metamathematics one is using to discuss things like whether a sentence in the internal theory is provable or not, and what constitutes a valid proof. The thing that is confusing for beginners is that both the internal and external language can have a concept of a natural number, but that these concepts can be distinct; in particular, it is possible to work with “nonstandard” models of the internal theory that contain additional “nonstandard” natural numbers that would not correspond to any natural number in the external metatheory.

When we define a proof to be a sequence of logical deductions of some finite length $m$ , the $m$ here is an external (standard) natural number, not an internal (possibly nonstandard) one. In particular, one cannot perform quantification over $m$ within the internal theory; only the external metamathematics.

Where things can get particularly confusing is when one starts formalizing notions of proof within the internal theory (in order to study things like the Godel incompleteness theorems). The internal theory can only reference the internal natural numbers, but not the external ones, so its encoding of the concept of a proof becomes subtly different. This leads to apparent “paradoxes” such as the existence of models of (say) Peano arithmetic which can “prove” their own inconsistency, but where their “proof” has a length that is a nonstandard natural number, rather than a standard one.

26 April, 2025 at 7:51 am

Guanyuming He

Dear Prof. Tao,

I think your footnote at p. 64, 4th ed. actually has a point to be mentioned again: if we use set theory language to formally define $a{--}b := \{ (c,d) \in \mathbf{N}\times\mathbf{N} \ \mid\ a+d = b+c \}$ , then we cannot also define $(a{--}b = c{--}d) := (a+d=b+c)$ as we did in Defn 4.1.1. Instead, we must prove it as a lemma, and I think it makes a good addition to the exercises of Section 4.1.

Thank you,
Guanyuming He

[Suggested exercises added – T.]

27 April, 2025 at 7:13 pm

Mr.Park

Sir, I’m curious the Exercise 11.10.4 in the version of Φ such that neither monotone increasing or monotone decreasing.

My try is like below: Let Φ:[a,b]->R be differentiable on [a,b].
Then Φ is continuous function, so by maximum principle(proposition 9.6.7) we can find x_max / x_min.
Then We can set Φ:[a,b]->[Φ(x_min),Φ(x_max)].
And suppose Φ’ is Riemann integrable.
And let f:[Φ(x_min),Φ(x_max)]->R be a Riemann integrable function on [Φ(x_min),Φ(x_max)].
Then the function (f∘Φ)Φ’:[a,b]->R is Riemann integrable on [a,b] and ∫[a,b](f∘Φ)Φ’=∫[f(x_min⁡),f(x_max⁡)]f.
I’m not sure whether this correction is right or not in the version
of Φ such that neither monotone increasing or monotone decreasing.
And if this correction is right, i wanna know the idea for solving this question and wonder how to solve it.

Thank you,sir.

5 May, 2025 at 10:15 am

Anonymous

About two-thirds of the way down page 63, there appears to be “defn” where you want “definition”.

I can’t confirm that this typo occurs in the very latest version of (edition 4) of the e-book because my institution seems to have lost access to that e-book. :-(

5 May, 2025 at 1:41 pm

Anonymous

More than once, the book uses the spelling “supercede”, which I think is deprecated in favor of “supersede” (regardless of the branch of English one prefers).

10 May, 2025 at 9:12 am

valiantb906db6cfd

Professor, I am studying about sets in Section 3.

The textbook states that for a set X, we can determine whether any mathematical object is included in X or not.

In probability theory, the sample space is treated as a set. Generally, when the sample space is a familiar set, such as the set of real numbers or natural numbers, it is not hard to accept.

What I am curious about is the following:

In the case of a coin toss, we can think of the sample space as {H, T}.

In this case, are H and T considered mathematical objects?

And if the answer to the above question is yes, does this mean that the concept of Type mentioned in Appendix A.7, regarding mathematical objects, suggests that we can handle the sample space using the concept of Type theory?

10 May, 2025 at 10:12 am

Terence Tao

Yes, in the orthodox foundations of probability theory, sample spaces are modeled as sets of mathematical objects, though one may wish to keep this mathematical model distinct from the real world system it is meant to describe. See also https://terrytao.wordpress.com/2015/09/29/275a-notes-0-foundations-of-probability-theory/

10 May, 2025 at 6:19 pm

valiantb906db6cfd

Thank you so much,sir :)

11 May, 2025 at 4:45 am

Guanyuming He

Dear Prof. Tao,

The manual of the proof assistant that I’m learning, Coq, states that the induction principles generated for inductively defined types relies on the principle of structural induction.

I followed into the Wikipedia page of structural induction, which claims that it’s equivalent to the well-ordering principle, and thus also equivalent to the standard mathematical induction. But it does not explain why. I wish to ask

Is the structural induction somehow another way of stating the “Strong induction” (Proposition 8.5.10) in your book? If I treat the inductively defined thing as a tree and impose an order on it, then it seems plausible. But I struggle to show that it’s a total ordering: we can’t always compare things on different branches.
And how to prove the principle of structural induction is equivalent to that of well-ordering, and of standard induction? Since inductively defined types must have a finite number of constructors, each branch of the tree can’t be infinite, then, it seems possible to prove that the number of nodes of the tree is countable, thus doing standard induction on the tree seems possible. Am I on the right path?

Thank you,
Guanyuming He

11 May, 2025 at 6:33 am

Terence Tao

Structural induction is most closely tied to well-ordering principle for well-founded sets, which is more general than the well-ordering principle for well-ordered sets as it does not require total ordering. So it is relatively easy to use structural induction to recover the ordinary induction principle, but going the other way is trickier (for instance, one can proceed here using Zorn’s lemma, which is one of the standard ways to obtain results about partially ordered sets from results about totally ordered sets).

27 May, 2025 at 1:32 pm

Guanyuming He

Dear Prof. Tao,

By using Zorn’s lemma, it seems that the proof doesn’t involve the standard induction at all. So, by “equivalent to structural induction”, does Wikipedia only mean when the set is countable?

BTW, Here’s my proof using Zorn’s lemma. If you have time and are willing, then I would be very grateful if you don’t mind checking if it’s correct.

Let $(X, \prec)$ be a well-founded set. Goal: Given: for any $x : X$ , if $P$ holds for all $y \prec x$ , then P holds for x. We have, P holds for all $x : X$ .

Define $\mathcal{F}$ as the set of all subsets of $X$ each of which includes all elements smaller than any of its elements: $\mathcal{F}= \{A \subseteq X \mid \forall x:A, y:X,\ y \prec x \to y \in A \}$

Then, use axiom of replacement to filter out those in $\mathcal{F}$ that
$P$ holds for all of the elements: $\mathcal{G} = \{ A \in \mathcal{F} \mid \forall x : A, P(x) \}$

Clearly, $\mathcal{G}$ is partially ordered by the subset relation. I
plan to apply Zorn’s lemma on it. To do so, I need to prove every
totally-ordered subset of it has an upper bound in $\mathcal{G}$ . Let
$I$ be any totally ordered subset of it. Take $A = \bigcup_{i \in I} i$ . Let $a \in A$ , then for some $i$ , $a \in i$. Thus,
because $i$ ‘s property, we have $P(x)$ and $y \prec x \to y \in i \to y \in A$ . This means $A \subseteq \mathcal{G}$ .
Clearly, $A$ is an upper bound of $I$ , and we can apply Zorn’s lemma
to $\mathcal{G}$ to obtain an maximal element $latex M \in
\mathcal{G}$.

To prove our goal, we need to show that $M = X$ . Suppose for the sake of
contradiction that $N:= X - M$ is non-empty. Because $X$ is
well-founded, we have a minimal $n \in N$ , which means $x:X \prec n \to x in M$ . But $x \in M \to P(x)$ , we thus have $x \prec n \to P(x)$ . By our hypothesis, we thus get $P(n)$ and $M+\{n\} \in mathcal{G}$ , contradicting to the fact that $M$ is the maximal
element.

Thank you,
Guanyuming He

28 May, 2025 at 12:29 pm

Terence Tao

This would be an excellent opportunity for you to use as an exercise to learn a formal proof assistant language such as Lean.

27 May, 2025 at 11:15 pm

Guanyuming He

A typo in the proof: axiom of replacement should be of specification. Also, I found I should have assumed $X$ is non-empty and then prove $\mathcal{G}$ is non-empty, finally adding non-empty to $I$ to apply Zorn’s lemma.

Otherwise my proof seems fine to me after three reviews. Don’t know if it looks good in your eyes, Prof. Tao.

27 May, 2025 at 11:42 pm

Guanyuming He

Also, the $x$ should have been $a$ when talking about $a \in A$, sorry.

20 May, 2025 at 2:10 pm

Anonymous

In Exercise 5.3.3, “equivalent” should be “are equivalent”.

[Erratum added, thanks – T.]

21 May, 2025 at 3:27 pm

Anonymous

Page 97: Last line should specify that the Sect 1.2 referenced is in Analysis II.

[Erratum added, thanks – T.]

25 May, 2025 at 7:22 am

Guanyuming He

Dear all,

If you are also interested in the foundation of mathematics after learning Prof. Tao’s Analysis I, I recommend this book that I found recently and have been reading: Kleene’s Introduction to Metamathematics.

Its first few two chapters have a large intersection with Analysis I’s foundational chapters (2, 3, and 8), but they were given in a different manner, describing how historically they were considered. For example, that book’s chapter 1 mainly focus on how Cantor studied abstract sets and cardinal numbers. Later, the book dives deep into mathematical logic. Then, it discusses in detail computability related topics, using recursive functions as the primary tool.

I found this book very helpful. I hope you find it helpful, too.

Yours truly,
Guanyuming He

31 May, 2025 at 12:33 am

valiantb906db6cfd

I have some additional questions I’d like to ask.

(Additional Questions)

(Question 1)

In A.7, you mentioned that the meaning of “equality” is essentially a matter of definition.

For example, let’s consider some type or class T.

Maybe we can imagine various kind of “definition candidates” for equality on a single type or type T.

Does this mean that regardless of how it is defined, as long as it satisfies the four axioms (1, 2, 3, 4), we can define countless different versions of “=” on a single type or class T based on different “definition candidates”?

(Question 2) In A.7, the terms type and class are used interchangeably when referring to T.

I’m curious about what exactly a type and a class are, respectively, and whether they refer to the same thing or not.

31 May, 2025 at 7:15 am

Terence Tao

A good formal setup for the concepts discussed in this appendix (which, as stressed at the start of that appendix, should not be read at a purely formal level) is that of type theory. Many modern formal proof assistant languages, such as Lean, are based on type theory (or on a more sophisticated version known as dependent type theory, which I won’t go into here). In this theory, all mathematical objects are assigned a type: some objects may be natural numbers, others may be integers, yet others may be functions or points in a Euclidean plane, and so forth. For instance, in Lean, the natural number 3 is denoted (3:Nat), while the integer 3 is denoted (3:Int) Technically, objects from different types are considered different objects: for instance (3:Nat) is not equal to (3:Int). However, there are “coercions” that allow one to convert an object of one type to another. For instance there is a coercion from the natural numbers to the integers that assigns to each natural number n its corresponding integer (n:Int), for instance (3:Nat) is assigned ((3:Nat):Int) = (3:Int).

Each type carries its own version of equality. But if one wants to impose a new notion of equality on a type, then (as long as it obeys the reflexive, symmetric, and transitive axioms) one can define a quotient type that is similar to the parent type, but with the new notion of equality imposed. Technically this is a different type from the original type, but there is a coercion from the latter to the former. For instance, in my text, the integers are essentially a quotient type of formal differences $a -\!- b$ of natural numbers, where the notion of equality used to perform the quotient is $a -\!- b = c -\!-d \iff a+d = c+b$ .

These distinctions are easiest to explore within the framework of a formal language such as Lean. I am in fact working right now on providing a companion to selected chapters of my textbook in the Lean language to assist in this: stay tuned!

31 May, 2025 at 8:49 am

A Lean companion to “Analysis I” | What's new

[…] 20 years ago, I wrote a textbook in real analysis called “Analysis I“. It was intended to complement the many good available analysis textbooks out there by […]

9 June, 2025 at 3:58 am

Anonymous

In light of the existing erratum for Lemma 11.1.4, should we also delete the word “non-empty” from the first sentence of the lemma?

[Erratum adjusted – T.]

9 June, 2025 at 4:15 am

Anonymous

In Exercise 11.1.3, I’m actually not convinced that it must be $c<b$ instead of $c\leq b$ . If $c=b$ , and one of the intervals is of the form $(c,b)$ or $[c,b)$ , then it equals the empty set. This still allows us to pull it out of the partition $\mathbf{P}$ and apply the induction hypothesis. Or am I missing something?

[The $c \leq b$ version of the exercise is also a true statement, but I prefer to state the stronger version here, as is done in the most recent erratum. -T.]

18 June, 2025 at 4:07 am

Anonymous

Hi Professor Tao,

In Lemma 7.5.2 on Page 156, we have this section of a proof relating to the ratio test:

$\displaystyle \text{This implies that } c_{n+1} \leq c_n (L + \varepsilon) \text{ for all } n \geq N$ .
$\displaystyle \text{By induction, this implies that } c_n \leq c_N (L + \varepsilon)^{n - N} \text{ for all } n \geq N (\text{why?}).$
$\displaystyle \text{If we write } A := c_N (L + \varepsilon)^{-N}, \text{ then we have } c_n \leq A (L + \varepsilon)^n$ .

However, I believe it must instead be $A := c_N (L + \varepsilon)^{N}$

[I believe the text is correct as stated – T.]

19 June, 2025 at 11:41 pm

Anonymous

I see my confusion very clearly now; I had been struggling to understand that proof for some time that day. I apologize! Thank you for the clarification.

23 June, 2025 at 1:23 pm

Anonymous

Dear Professor Tao,

I wasn’t understanding why we have to go to such great lengths to prove Lemma 3.5.11 (Finite Choice) because, using the same reasoning as in the base case $n=1$ in its proof, I’m seeing that Lemma 3.5.11 follows instantly. In the base case $n = 1$ , we just said that because there exists an object $a \in X_1$ , then we can “choose” it by defining $x_1:= a$ . So for any natural number $n \ge 1$ , what is preventing us from saying that for all $1 \le i \le n$ , there exists an object $a_i \in X_i$ (by Lemma 3.1.5 [Single Choice]) that we can “choose” by defining $x_i = a_i$ , and this sufficing as our proof of Lemma 3.5.11? Am I misunderstanding the argument used in the base case $n = 1$ ?

24 June, 2025 at 9:37 am

Terence Tao

This arises from a subtle distinction between the “internal” natural numbers in one’s mathematical theory (which, in this text, are coming from Assumption 2.6 or Axiom 3.8), and the “external” natural numbers that are present in the ambient metamathematics. In the metamathematics, a proof is only acceptable if it has a finite length, where “finite” means “having a length equal to an external natural number”. For instance, a proof that is 100 lines long would be acceptable, and so one can iterate single choice to generate 100 choices for instance. But it is theoretically possible that the internal natural numbers contain “nonstandard numbers” which do not correspond to any external natural number such as 100. As such, it is not always possible to validly create proofs involving $n$ steps for a given internal natural number $n$ . However, the principle of mathematical induction still allows one to establish statements involving such numbers (with proofs that are externally finite).

Incidentally, Lemma 3.5.11 is now formalized in Lean at https://teorth.github.io/analysis/docs/Analysis/Section_3_5.html#Chapter3.SetTheory.Set.finite_choice (or https://github.com/teorth/analysis/blob/933737363f0dcc7153b61e8e273d1e6993e10d1a/analysis/Analysis/Section_3_5.lean#L278 ). Lean does not directly offer the ability to perform `n` tasks for any (internal) natural number `n`, so one must proceed by induction instead.

24 June, 2025 at 12:41 pm

Guanyuming He

Dear Prof. Tao,

Since classical mathematics has many results that do things beyond the finitary methods as used in the meta theory anyway, must proofs in the object theory always be finitary? Is the problem here that we must be able to study proof as objects, and if proofs are not finite in our informal sense, it would not be possible?

Thank you,
Guanyuming He

25 June, 2025 at 12:53 pm

Terence Tao

There are infinitary logics that allow for the possibility of infinitely long proofs, but by default, mathematics largely works within the metamathematical framework of first-order logic, in which proofs must be of an (externally) finite length. Infinitary logic is a rather subtle topic and not one I would recommend for beginners, or for students not specializing in mathematical logic.

24 June, 2025 at 1:41 pm

Anonymous

Thank you; I thought in our framework that all models of the natural numbers are isomorphic to the standard model of natural numbers in view of exercise 3.5.13? Online I read that nonstandard models aren’t isomorphic to this. Am I misunderstanding what exercise 3.5.13 is proving?

25 June, 2025 at 12:48 pm

Terence Tao

Exercise 3.5.13 shows that all internal models of the natural numbers within the mathematical theory (in this case, ZF set theory) are (internally) isomorphic. However, the external natural numbers that are used by the ambient metamathematics need not be isomorphic to the internal models.

An analogy is with computers that have internal representations of numbers, for instance as 16-bit or 32-bit strings. They can be standardized in such a way that any two internal data types of integers can be mapped isomorphically to each other, but they may not necessarily correspond to what external observers would think of as numbers; in particular, computer representations of numbers can be subject to overflow errors that the external number system would have.

25 June, 2025 at 10:26 pm

Anonymous

How would I reconcile “the internal natural numbers contain “nonstandard numbers” which do not correspond to any external natural number” and “all internal models of the natural numbers within the mathematical theory (in this case, ZF set theory) are (internally) isomorphic”? Because I thought the latter quote implies that the internal models of the naturals are internally standard, as they are isomorphic to the standard natural numbers?

27 June, 2025 at 11:04 am

Terence Tao

There are multiple models of ZF set theory. Each one in turn contains models of the natural numbers, which are internally isomorphic to each other; but there are nonstandard models of ZF set theory which would contain nonstandard models of arithmetic. But within any such model of set theory, all the models of arithmetic would be isomorphic to each other.

26 June, 2025 at 5:06 pm

Anonymous

For exercise 3.5.12, don’t we need to have $\mathbb{N} \subseteq X$ as one of our assumptions? As otherwise $a(n)$ for some $n \in N$ may not be defined?

[As per the errata, $a(n)$ should take values in $X$ rather than ${\bf N}$ . -T]

29 June, 2025 at 3:41 pm

Anonymous

Exercise 11.6.5 feels odd. We are asked to use the integral test to prove Corollary 11.6.5. However, the integral test is only applicable for $p\geq 0$ , and we already have Corollary 7.3.7 when $p>0$ . So at best we can cover an edge case with the integral test, but then we need something else (like the zero test) to complete the proof.

Furthermore, we are allowed to use the second fundamental theorem of calculus, but then to apply this in the case $p=1$ , we would need to know about the natural logarithm function, which we don’t have at this stage.

Of course we can always prove a particular result in multiple ways, but I feel like a different example problem to apply the integral test would maybe be better.

30 June, 2025 at 9:01 am

Terence Tao

Fair enough; I think I will delete Corollary 11.6.5 and move Exercise 11.6.5 to an exercise in Section 11.9 instead, asking to provide an alternate proof of Corollary 7.3.7.

3 July, 2025 at 3:45 am

Anonymous

Professor Tao, I have found Section 11.8 rather difficult to follow. There are certain assumptions on $\alpha$ that slip in and out of focus and it’s hard to tell what restrictions apply where. I have tried to summarise below.

Initially, we define $\alpha$ -length assuming that $\alpha$ is monotone increasing. Then we note that this allows for a simpler formula when $\alpha$ is continuous but not necessarily monotone increasing.
We prove Lemma 11.8.4 assuming that $\alpha$ is monotone increasing OR continuous. We then use the same assumption in Definition 11.8.5.
We prove an analogue of Proposition 11.2.13 assuming that $\alpha$ is monotone increasing OR continuous. The text then says we can define the piecewise-constant Riemann-Stieltjes integral for any piecewise constant $f$ and “any” $\alpha$ , but surely this is too broad.
We now bring back the strict assumption that $\alpha$ is monotone increasing before we can prove an analogue of Theorem 11.2.6. Is this only because parts (d) and (e) aren’t necessarily true otherwise?
Is this strict assumption dropped again when defining the full Riemann-Stieltjes integral? Is it not required for proving the analogue of Theorem 11.5.1? Again, the paragraph defining upper and lower integrals doesn’t place any restrictions on $\alpha$ apart from being defined on a closed interval containing $I$ .

[I plan to formalize this section in Lean in the near future, at which point I should be able to clarify the situation. -T]

4 July, 2025 at 8:25 pm

Terence Tao

I have just now formalized this section. It ended up that it was natural to define $\alpha$ -length in terms of right limits and left limits, so the natural range here is for those functions for which left and right limits always exist; if the functions are defined only on an interval, they should be extended in such a fashion that one has right continuity at the right endpoint (if this is in the domain) and similarly at the left. This covers both the monotone and continuous cases.

Piecewise constant Riemann-Stieltjes integrals are defined (and reasonably well-behaved) in this context, but for most of the later results regarding this integral, monotonicity will be required (and this is implicit starting from the sentence “Let us now assume that $\alpha$ is monotone increasing” in this section. Hopefully as the Lean exercises get attempted, it will become clearer precisely which portions of Theorem 11.2.6 actually require monotonicity; I can imagine in some cases that one can get by without this hypothesis.

5 July, 2025 at 3:55 am

Anonymous

Many thanks, Professor Tao.

12 July, 2025 at 8:18 pm

Anonymous

Does these errata appear corrected in the latest printings? Or it is incorporated only in future editions of the book?

[The latter -T.]

15 July, 2025 at 1:08 pm

Anonymous

Not sure if this is this was intentional and if anyone else cares, but I’ve noticed that the font in the Fourth edition PDF changed to something that looks very closely to Times New Roman — I thought at first that my PDF renderer was not working correctly. I think this has hurt the aesthetic charm of the book, as this change made it feel like an overly long Word document. Maybe I should send this feedback to the publisher.

(Old on the left; new on the right)

17 July, 2025 at 7:53 pm

Anonymous

Jumping in here to add an erratum I just spotted. Example 3.4.3 says that the integers will be rigorously defined in the next section, but it’s actually in the next chapter. :)

[Erratum added, thanks – T.]

17 July, 2025 at 10:46 pm

Anonymous

On page 293, in Exercise B.2.3, the statement is actually false. Counterexample: pick $x=0=0/10^0$ , so $x$ is a terminating decimal, but it has only one decimal representation.

19 July, 2025 at 2:42 pm

Terence Tao

In this appendix we consider $+0.00\dots$ and $-0.00\dots$ to be distinct decimal representations of the same real number $0$ .

30 July, 2025 at 8:03 am

valiantb906db6cfd

I have some additional questions about “appendix A.7”.

Question 1:
Suppose we introduce a new type for which equality has not yet been defined. In that case, we would need to define equality. According to the four axioms presented in A.7, can we say that there is only one definition of equality that satisfies them? Or could there be multiple definitions that satisfy those axioms? If multiple definitions are possible, how should we decide which definition of equality to choose?

Question 2:
Is it possible to formulate the concept of equality without using type theory?

1 August, 2025 at 7:11 pm

Terence Tao

In type theory, one can implement equality in the spirit of Appendix A.7 using the concept of quotient types https://en.wikipedia.org/wiki/Quotient_type ; in Lean, this is formalized as Quot.sound, one of the three basic axioms of Lean’s mathematical library (the other two are propositional extensionality and the axiom of choice). Setting a different notion of equality would lead to a different quotient type. For instance, the integers quotiented by the equivalence relation of having the same final digit base 10 would lead to a type that is essentially the cyclic group ${\mathbf Z}/10{\mathbf Z}$ , but using instead base 12 instead of base 10 to define the equivalence relation would lead to a different quotient type ${\mathbf Z}/12{\mathbf Z}$ .

In the more orthodox formulation of first-order set theory, one does not directly have the ability to redefine equality in this fashion, although one can still create quotient spaces as sets of equivalence classes of an equivalence relation. At a metamathematical level, one can also introduce new sorts (the analogue of types) to a first-order theory through the mechanism of conservative extensions, which can also be used to create analogues of quotient spaces or quotient types even without an internal set theory, but this requires a certain amount of proof theory and/or model theory to set up properly.

12 September, 2025 at 10:40 pm

seeker

Hi Dr Tao,

I had a question regarding the two different definitions of a partition and the relationship of these to the continuity of $\alpha$ in the Riemann Stieljtes integral.

A partition of a bounded interval $I$ in the book is defined as the set $P$ of intervals such that the intersection of any distinct intervals is empty and the union of all intervals is $I$ .

Another common definition of a partition of an interval $I$ is a list of points $a=x_0 \le \dots \le x_n=b$ .

I was attempting the following exercise from PMA by Walter Rudin and noticed that if we use the first definition of a partition, then we don’t need $\alpha$ to be continuous. But if we go with the second one, then we do.

Exercise: Suppose $\alpha$ increases on ${}[a, b]$ , $a\le x_0\le b$ , $\alpha$ is continuous at $x_0$ , $f(x_0) = 1$ , and $f(x) = 0$ for $x\ne x_0$ . Prove that $f$ is integrable and $\int_{[a, b]}f d\alpha = 0$ .

I know that the book doesn’t define the upper Riemann-Stieltjes sums but if we were to define them similarly to the Riemann sums, does the upper and lower Riemann-Stieltjes integral equal the infimum and supremum of the upper and lower Riemann-Stieltjes sums?

Using the definition of partitions given in the book, the infimum of the upper Riemann-Stieltjes sums is attained when the partition $P$ contains $x_0$ as a singleton set. This singleton set is still an interval. For any other partition that contains $x_0$ in an interval $J$ that is not a singleton set, the upper sum is $\alpha[J]\ge 0$ . So this isn’t the infimum. This way, we never need to reason about the continuity of $\alpha$ .

But if we use the second definition of a partition, then I can see the need for continuity as we obtain the infimum by making the points in the list as close as possible and continuity gives us that the length of the interval containing $x_0$ reaches $0$ as $\alpha$ gets close to $\alpha(x_0)$ for points in $(x_0 - \delta, x_0 + \delta)$ .

Am I thinking about this the wrong way or is continuity not needed if one chooses the first definition of a partition? Is there anyway to show their equivalence?

13 September, 2025 at 6:12 pm

Terence Tao

With the definitions in my book (together with the errata), if $\alpha$ has a jump discontinuity at $x_0$ , then the singleton $\{x_0\}$ can have a positive $\alpha$ -length, and the Riemann-Stieltjes integral will be positive in this example. I believe the two notions of integral should coincide assuming right-continuity of $\alpha$ (but I don’t have Rudin’s book at hand to confirm for sure).

You may also be interested in the formalization of this section of the text in Lean at https://teorth.github.io/analysis/docs/Analysis/Section_11_8.html

18 October, 2025 at 11:20 pm

Noel Hinton

I have just finished your excellent textbook and I’m looking forward to starting Analysis II. I have a list of possible corrections for the tail end of the book.

In the hint for Exercise 9.1.4, “axiom of choice” should be “the axiom of choice”.
In Exercise 9.4.1, “(a),(b)” needs a space.
In Exercise 9.6.1(c), the period at the end should be a semicolon.
In the hint for Exercise 9.8.5(c), I think that the definition for the functions $f_n$ should be written $f_n(y):=\dots$ , because x is already assumed to be a particular irrational number.
In the first paragraph of Section 9.9, I think “continuous at $(0,2)$ ” should be “continuous on $(0,2)$ “.
Most of the labels attached to definitions and results have a consistent formatting. They are usually enclosed in parentheses, and the definition labels are italicised. There are a few labels that don’t follow this convention, mostly towards the end of the book. E.g., Definition 9.9.2, Theorem 10.1.3, Theorem 10.1.15, Theorem 10.2.7, Theorem 10.4.2, and Lemma 11.10.5.
On page 212, prior to the revised definition of equivalent sequences (Definition 9.9.5, superseding Definition 5.2.6), there is a note that says we no longer require the sequences to be Cauchy. But Definition 5.2.6 doesn’t require the sequences to be Cauchy either; as written, it assumes two arbitrary sequences of rational numbers.
In the proof of Theorem 9.9.16, the reference to Proposition 8.1.5 is not hyperlinked.
There are several instances of $\inf$ in Chapter 9 that probably should be $+\inf$ according to the convention you have adopted elsewhere in the text. (That is, you define the extended real numbers $+\inf$ and $-\inf$ , not $\inf$ and $-\inf$ .) See Lemma 9.1.12, Example 9.4.6, Proposition 9.4.11, Definition 9.5.1, Proposition 9.5.3, Exercise 9.6.1(b), Example 9.8.2, and Example 9.10.4.
Every proof in Chapter 10 is missing an end-of-proof square. (Maybe this is one of those odd issues that only affect certain versions of the digital text.)
In the last sentence above Proposition 10.3.3, “if function” should be “if a function”.
The hint for Exercise 11.1.2 is slightly incorrect (unless I’m missing something). It is true that the intersection of two connected sets is bounded, but not necessarily true that the intersection of two connected sets is connected. The intersection may be empty, and connected sets by definition are non-empty.
In Proposition 11.10.1, the symbol $I$ is introduced to denote the interval $[a,b]$ , but then is hardly used.
When writing integrals, the text sometimes has $\mathrm{d}x$ , and sometimes has $dx$ . (Just using $x$ as an example.) You can see both styles on pages 260-262.

Many thanks.

Noel

[Added, thanks – T.]

12 November, 2025 at 7:24 am

Anonymous

Prof. Tao,

Geometrically, the definite integral can be viewed as the area under the curve. But what is the geometric description of the indefinite integral?

16 November, 2025 at 7:37 am

Anonymous

Prof. Tao,
The technique of Taylor expansion arises quite a lot in analysis, such as solving some types of limits problems in undergrad. analysis. Can you give us a general problem-solving strategy of when to use this technique?

22 November, 2025 at 12:08 am

Noel Hinton

Professor Tao, I have an idea for a bonus exercise in (or after) Section 3.5; it’s about an alternative definition for ordering the natural numbers.

Let $m$ and $n$ be natural numbers. Let $A_m$ and $A_n$ be their corresponding sets from Exercise 3.5.12(b). We write $m\leq n$ iff $A_m \subseteq A_n$ . Using this new definition, prove the properties of order in Propositions 2.2.12 and 2.2.13, and prove that this definition is equivalent to Definition 2.2.11.

27 November, 2025 at 6:16 pm

valiantb906db6cfd

Professor, I have a question.In Volume 1, you explained ZFC set theory. As I understand it, ZFC represents all mathematical objects as sets.You also mentioned that equality of sets is expressed as an axiom in ZFC. This seems to assume that sets already satisfy the four basic properties of equality.Of course, reflexivity, symmetry, and transitivity can be easily verified.

Originally, one would have to check whether the substitution axiom holds, but since ZFC already states equality as an axiom, it follows that the substitution axiom is satisfied. Therefore, any operation f or predicate P that would violate substitution is simply not definable.

In Appendix A.7, equality is explained from the perspective of type theory. If it can be explained in type theory, I think it should also be explainable within ZFC set theory. I would like to understand in more detail how equality is handled in set theory.

For example, in the earlier sections, when constructing integers, rationals, and reals, you showed their existence by mapping each object to a set. In this case, we identified specific objects with parts of other objects.On the other hand, in Section 8.4, it is stated that two finite Cartesian product sets have a “canonical bijection,” and can thus be considered the same set. I am not sure why, but I suspect it is related to the equality discussed in Appendix A.7. However, why is it permissible to regard these two sets as equal? The elements of one set are functions, while those of the other are n-tuples, so strictly speaking the objects are different.

30 November, 2025 at 10:43 am

Terence Tao

Orthodox set theory is formulated within the framework of first-order logic. In this logic, one has to specify in advance a collection of “sorts” (e.g., the “sort” of sets, or the “sort” of natural numbers): for simplicitly one can work with pure set theory, which is one-sorted in that the only objects under consideration are sets. As part of the axioms of first-order logic, an equality operation on each sort is assumed to exist and obey the laws of equality listed in Appendix A.7. However, unlike type theory, no creation of new sorts are permitted within this framework. Of course, in practice we do want to create new sorts, since it would be extremely limiting to have sets as the only possible sort of mathematical object one can refer to; but in this framework the only way to do this is to work at the metamathematical level and create various conservative extensions of the original base theory that introduces new sorts. This can be understood reasonably well on an informal level, but is rather awkward to implement formally; this is one major reason why most current formal proof assistants use some form of type theory rather than first-order logic. In particular, the Lean formalization of this textbook uses type theory.

Technically, identifying two types that are canonically isomorphic is an abuse of notation, and indeed when formalizing these concepts carefully (as is done in the above repository) one has to keep such types separate (although one can still effectively perform something close to an identification by using implicit coercions, for instance coercing natural numbers to be integers, integers to be reals, etc.). But such abuses of notation are very convenient for informal mathematical reasoning, even if occasionally they trip up human mathematicians with their subtleties. For further discussion of these topics, I recommend the talk “Formalizing invisible mathematics” by Andrej Bauer.

1 December, 2025 at 6:47 pm

Anonymous

It is said in Remark 2.1.14 that “our definition of the natural numbers is axiomatic rather than constructive.”

It is understandable that Peano axioms alone are axiomatic. But it seems that Definition 2.1.3 does provide a construction: one begins with the primitive symbol 0 and then define 1:=0++, 2:=(0++)++, etc.

How should one understand the nuance here?

1 December, 2025 at 7:22 pm

Terence Tao

In this setup, the notation for $1$ , $2$ , etc. is strictly speaking not part of the core natural number system (which are described by the Peano axioms), but is an additional constructed layer of notation on top of the natural numbers.

1 December, 2025 at 8:32 pm

Anonymous

Prof. Tao,

you define “a set is a collection of objects”, the word “collection” here in this definition is the natural language right? I am kind of confused because I thought that everything in mathematics would be formalised ( or defined). And I actually don’t understand what formal language is and how they work in mathematics.

1 December, 2025 at 8:35 pm

Terence Tao

Mathematics contains both formal and informal components; for instance, Definition 3.1.1 is explicitly marked as informal. See https://terrytao.wordpress.com/career-advice/theres-more-to-mathematics-than-rigour-and-proofs/

If you would prefer a completely formal version of this text, you can check out the Lean version of the text at https://github.com/teorth/analysis . For instance, the definition of a set is contained in https://github.com/teorth/analysis/blob/main/analysis/Analysis/Section_3_1.lean

1 December, 2025 at 9:00 pm

Anonymous

Could you please explain what “formal language” is?

[ https://en.wikipedia.org/wiki/Formal_language ]

4 December, 2025 at 6:33 pm

Anonymous

Could you explain briefly why Axiom 3.2 must be an “axiom” instead of a “definition”?

Isn’t it simply using the more previous notion “every element x of A belongs also to B and every element x of B belongs also to A” to “define” what the new notion “A=B”?

Does this have anything to do with the fact that “=” is already used in Appendix A7 so one needs to “reload” the symbol somehow? (Though that does not seem to explain why it can be a “definition”.)

It would be helpful if one can contrast “axiom” vs “definition” using examples in Chapter 2.

6 December, 2025 at 8:36 am

Terence Tao

In the informal type theory used in this text, one can impose a notion of equality A=B on a type of objects, so long as all the operations one has defined on this type obey the axioms of equality in Appendix A7. For instance, when defining the integers as formal differences $a -\!\!\!- b$ of natural numbers, the definition of equality provided in Definition 4.1.1 is compatible with operations such as the definition of addition in Definition 4.1.2 because the law of substitution is respected (see also the Lean formalization of these constructions using the `Quotient` type).

For set-theoretic equality, it is *almost* possible to take Axiom 3.2 as a definition: the axioms of reflexivity, symmetry, and transitivity are obeyed by this definition, and the membership relation $x \in A$ is preserved by replacing the set $A$ with an equal set $B$ , giving one of the two cases needed for the axiom of substitution for this relation. However, as was pointed out to me by previous commenters, the object $x$ could also be a set, and if $x$ is equal to $y$ by the definition of equality of sets provided, it is not clear that the assertions $x \in A$ and $y \in A$ are logically equivalent. So the full substitution axiom is not verifiable directly from the definition, and must instead be taken as an axiom (which is what we do also in the Lean formalization).

6 December, 2025 at 10:15 pm

Anonymous

Dr Tao,

So in general what is the difference between “axioms” and “definitions”?

[For linguistic questions of this nature, I suggest directing your query to web search engines or AI chatbots. -T]

6 December, 2025 at 12:05 pm

Okay, I think I understand this better now. One should really view Axiom 3.2 as consisting of two logically distinct parts. Let $A$ and $B$ be sets.

(1) If $A=B$ , then every element of $A$ is an element of $B$ and vice versa. This direction is essentially an application of the substitution axiom for the symbol “ $=$ ”. By Axiom 3.1, sets are objects, and for any fixed object $x$ , the statement $P(z):= (x \in z)$ is a well-formed formula. Since “ $=$ ” is a primitive symbol in the underlying first-order logic and satisfies the equality axioms in Appendix A7, we automatically have
$\displaystyle A=B \;\Rightarrow\; (x\in A \leftrightarrow\ x\in B)$
for all $x$ .

(2) Conversely, if every element of $A$ is an element of $B$ and vice versa, then $A=B$ . This is the genuine content of the extensionality axiom in formal ZFC: it tells us when $A=B$ , while (1) merely describes what follows once $A=B$ .

Seen this way, Axiom 3.2 is not a definition of “ $=$ ”. The symbol “ $=$ ” is already part of the underlying first-order logic and comes equipped with the equality axioms in Appendix A7. Axiom 3.2 merely constrains equality for the particular case of sets.

If instead one works in first-order logic without equality, then attempting to define $A=B$ purely as “every element of $A$ is an element of $B$ and vice versa” runs into exactly the substitution problem you mentioned. As noted in the Wikipedia article on ZFC, one possible way around this (in a no-equality framework) is to define
$\displaystyle A = B \;\text{iff}\; (\forall x)(x\in A \leftrightarrow\ x\in B)\ \text{and}\ (\forall C)(A\in C \leftrightarrow\ B\in C)$ ,
which enforces substitution by definition.

(I am curious whether something like this could be used as a definition in your Lean formalization, rather than taking extensionality as an axiom.)

7 December, 2025 at 10:19 am

Terence Tao

Yes, this is an accurate summary. In Lean, one could set up a ZFC style set theory without equality and then use Lean’s Quotient type to enforce extensional equality along the lines you suggest, at which point the usual ZFC axioms will hold for the quotient type; you are welcome to try to work with this (perhaps using the Lean formalization of my text as a starting point) if you want to test out both your first-order logic skills and your Lean programming skills.

8 December, 2025 at 12:55 pm

Anonymous

General LaTeX issue: You should probably use colon from the mathtools package instead of : for maps. This gives the “correct” spacing.

[Noted for the next revision, thanks – T.]

14 December, 2025 at 11:21 am

In Proposition 4.4.4, could one shorten the argument by assuming from the beginning that the rational number is written in lowest terms, i.e. $x=p/q$ with $p$ and $q$ not both even? Then $p^2=2q^2$ immediately forces both $p$ and $q$ to be even, giving a contradiction without explicitly invoking the principle of infinite descent.

That said, I feel that if one tries to justify why we may assume $p$ and $q$ are not both even, the justification itself seems to rely on an infinite descent–type argument. Is there a way to avoid this, or are the two approaches essentially equivalent at a foundational level?

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