Last updated: July 17, 2021

Analysis, Volume I

Terence Tao

Hindustan Book Agency, January 2006. Third edition, 2014

Hardcover, 368 pages.ISBN 81-85931-62-3 (first 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 third edition.

There are no solution guides for this text.

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

— Errata —

- p. 2, item 3: “can you add” should be “Can you add”.
- p. 9, line 5: “right-hand side” should be “left-hand side”.
- p. 10, first display: should be .
- p. 5, line 6 from bottom: should be . (Actually, for pedagogical reasons, it may be slightly better to use throughout this example instead of .)
- p. 59, Lemma 3.3.12: f should map Z to W, and h should map X to Y. In the proof of this lemma (on page 60): is a function from X to Z, and is a function from Y to W.
- p. 67, last paragraph: should be .
- p. 98: In Exercise 4.2.1, Corollary 2.3.7 should be Corollary 4.1.9. In Exercise 4.2.6, should be rational numbers, not real.
- p. 101: In Definition 4.3.9, after ““, add “; in particular, we define “.
- p. 127: In Exercise 5.3.4: add “(Hint: use Exercise 5.2.2.)”.
- p. 131, line 12 from bottom: “they cannot be than” should be “they cannot be larger than”.
- p. 175, Exercise 6.6.3: In the hint, replace “introduce” by “recursively introduce”, and insert “; ” after “” (two occurrences), with the parenthetical “(omitting the condition when )” inserted after the recursive definition of .
- Page 181: In Lemma 7.1.4(c), a period is missing at the end of .
- p. 183: In the proof of Proposition 7.1.8, should be replaced by in every display of the proof in which it appears.
- p. 197, in second line of proof of Proposition 7.3.4: the second sum should be rather than .
- p. 216, Exercise 8.1.9: It needs to be noted that this exercise requires the axiom of choice from Section 8.4.
- p. 220, Lemma 8.2.5: It needs to be noted that this lemma requires the axiom of choice from Section 8.4. Similarly, the case in Proposition 8.2.6 in which X is uncountable requires the axiom of choice also.
- p. 227, Exercise 8.3.2: should be .
- p. 236, last line: “for any good set Y'” should be “for any good set Y’ with non-empty”.
- p. 250: In Definition 9.10.3, “there exists an ” should be “there exists a real number “. Also add “let be a real number” to the first sentence of the definition.
- p. 255, Proposition 9.3.9(b): should be .
- p. 303, Exercise 10.4.3(a): The limit should be in the set rather than .
- p. 336, line 13: replace “we have made no assumption on ” with “the function could have been arbitrary”.
- p. 337, Exercise 11.8.1: Lemma 11.8.1 should be Lemma 11.8.4.
- p. 337, Exercise 11.8.5: In the last display, should be .
- p. 342, Exercise 11.9.1: “the function f is not differentiable” should be “the function is not differentiable.
- p. 383, first display: should be .
- p. 387, fourth display: should be .

— Errata for the second edition (hardback) —

- p. xii, bottom: “solidifed” –> “solidified”.
- p. xiv, top: “to know how to to” –> “to know how to”.
- p. 19. In footnote 2, add: “In the converse direction, if we have , then we may deduce ; this is the axiom of substitution (see Appendix A.7) applied to the operation .”
- p. 24, after Definition 2.2.1: “defined for every integer ” should be “defined for every natural number “.
- p. 26, after Proposition 2.2.6: “these notes” should be “this text”.
- p. 28, Proposition 2.2.14: “and Let” should be “and let”.
- p. 30, Lemma 2.3.3: “Natural numbers have no zero divisors” should read “Positive natural numbers have no zero divisors”.
- p. 32, Definition 2.3.11: Add the remark “In particular, we define to equal .”
- p. 37, Example 3.1.10: “(why?)” should be “(why?))”.
- p. 45: “8-m, where n is a…” should be “8-m, where m is a…”. In Exercise 3.1.2, add Axiom 3.1 to the list of permitted axioms. In Exercise 3.1.1: (3.1.4) should be Definition 3.1.4.
- p. 50: In the first line, should be , and should be .
- p. 55, Exercise 3.3.1: and should be and respectively.
- p. 59: In Lemma 3.4.9, “Then the set … is a set” should read “Then there is a unique set of the form … . That is to say, there is a set such that for any , if and only if is a subset of .
- p. 61: In Exercise 3.4.8, Axiom 3.1 should be added to the list of permitted axioms.
- p. 64: In Example 3.5.9, “” should be ““.
- p. 70, 4th line of proof of Lemma 3.6.9: should be . In the 6th line of proof of Proposition 3.6.8: Proposition 3.6.4 should be Lemma 3.6.9. After Lemma 3.6.9, add the following remark: “Strictly speaking, the expression has not yet been defined. For the purposes of this lemma, we temporarily define it to be the unique natural number such that (which exists and is unique by Lemma 2.2.10).”
- p. 81, before Lemma 4.2.3: “product of a rational number” -> “product of two rational numbers”.
- p. 84, before Definition 4.2.6: a space is missing between “Proposition 4.2.4” and “allows”. Before this paragraph, add “In a similar spirit, we define subtraction on the rationals by the formula , just as we did with the integers.”
- p. 86: In Definition 4.3.2, “real numbers” should be “rational numbers”. In definition 4.3.4, “be a rational number” should be added after “Let “.
- p. 88: In Proposition 4.3.10(b), the hypothesis n>0 should be added.
- p. 104, proof of Lemma 5.3.7; after invoking Proposition 4.3.7, add “(extended in the obvious manner to the case)”.
- p. 105, after Proposition 5.3.10: should be .
- p. 108, proof of Lemma 5.3.15: should be . “This shows that ” should read “This shows that “.
- p. 115: In the hint for Exercise 5.4.8, add “or Corollary 5.4.10” after “use Proposition 5.4.9”.
- p. 120: Add an additional exercise, Exercise 5.5.5: “Establish an analogue of Proposition 5.4.14, in which “rational” is replaced by “irrational”.”
- p. 124, Exercise 5.6.3: Add the hypothesis that x is non-zero (since the roots of 0 are not yet defined).
- p. 126, proof of Proposition 6.1.4: Proposition 5.4.14 should be Proposition 5.4.12.
- p. 134: In Definition 6.2.6(c) (and also on the first line of p. 135), should be .
- p. 135, Theorem 6.2.11(b), (c): Replace “Suppose that ” with “Suppose that ” (two occurrences). Exercise 6.2.2: Proposition 6.2.11 should be Theorem 6.2.11.
- p.144: Cor. 6.4.14: line 4: ” .. for all ” should be ” .. for all “
- p.146: proof of Theorem 6.4.18: Replace “from Corollary 6.1.17” here by “from Lemma 5.1.15 (or more precisely, the extension of that lemma to the real numbers, which is proven in exactly the same fashion)”.
- p. 151, Exercise 6.6.5: Replace “the formula , explaining why the set is non-empty” with “the recursive formula , with the convention , explaining why the set is non-empty”.
- p. 164, Definition 7.2.2: should be .
- p. 169, Exercise 7.2.6: Add “How does the proposition change if we assume that does not converge to zero, but instead converges to some other real number ?”. After Corollary 7.3.2: “conditionally divergent” should be “not conditionally convergent”, similarly in Exercise 7.2.13.
- p. 176: “absolutely divergent series” should be “series that is not absolutely convergent”.
- p. 177, Theorem 7.5.1: “conditionally divergent” should be “not conditionally convergent”, and similarly “absolutely divergent” should be “not absolutely convergent”. Similarly for Corollary 7.5.3 on page 179.
- p. 186, Exercise 8.1.1: This exercise requires the axiom of choice, Axiom 8.1. In Exercise 8.1.4. should be .
- p. 192, proof of Theorem 8.2.8: “absolutely divergent” should be “not absolutely convergent” (two occurrences).
- p. 196, Remark 8.3.6: “Paul Cohen (1934-)” should now be “Paul Cohen (1934-2007)”. :-(
- p. 197, Exercise 8.3.2: should be an injection rather than a bijection. In the definition of , should be (two occurrences).
- p 198, Example 8.4.2: after “the same set”, add the parenthetical “(or more precisely, in one-to-one-correspondence with)”.
- p. 200, Exercise 8.4.1: should be .
- p. 206, Exercise 8.5.5: “” should be “ or “.
- p. 208, Exercise 8.5.19: should be . In Exercise 8.5.20, the additional hypothesis “Assume that does not contain the empty set ” should be added.
- p. 214, Lemma 9.1.21. One needs the additional hypothesis “We assume that .”
- p. 220, Definition 9.3.6: “ is -close to near ” should be “, after restricting to , is -close to near “.
- p. 228, Proposition 9.4.7: change “three items” to “four items”, and add “(d): For every , there exists a such that for all with .
- p. 232, proof of Proposition 9.5.3: after “Proposition 9.4.7”, add “(applied to the restriction of to the subdomain )”.
- p. 252, Proposition 10.1.7: One needs the additional hypothesis . Similarly for Proposition 10.1.10, Theorem 10.1.13, and Proposition 10.3.1.
- p. 253, Definition 10.1.11: “For every ” should be “For every limit point “.
- p. 254, Remark 10.1.14: Leibnitz should be Leibniz (two occurrences).
- p. 256, Exercise 10.1.1: “ is also limit point of ” should be “, and is also a limit point of “.
- p. 257, Definition 10.2.1: should be .
- p. 262: In the proof of Theorem 10.4.2,”” should be ““.
- p. 271, Remark 11.2.2: “constant on ” should be “constant on “.
- p. 290: In Exercise 11.6.5, add “For this exercise, you may use the second Fundamental Theorem of Calculus (Theorem 11.9.4); there is no circularity, because Corollary 11.6.5 is not used in the proof of that theorem.”
- p. 290: In the proof of Proposition 11.7.1, in the third display, should be .
- p. 299: In Exercise 11.9.1, the hint is misleading (it requires the mean value theorem for integrals rather than for derivatives, which is not covered in this text) and should be deleted.

— Errata to the third edition (hardback) —

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

— Errata to the corrected third edition (hardback) —

- Page 1: On the final line, 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 15: In Prpoosition 1.2.15, should be (two occurrences).
- 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 Praroposition 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 .
- Page 27: In the final sentence of Definition 2.2.7, the period should be inside the parentheses. In proposition 2.2.8, “ is positive” should be “ is a positive natural number”.
- Page 29: Add Exercise 2.2.8: “Let be a natural number, and let be a property pertaining to the natural numbers such that whenever is true, is true. Show that if is true, then is true for all . This principle is sometimes referred to as
*the principle of induction starting from the base case*“. - 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 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 will often be abbreviated to “the integers”.”
- Page 47: In “In did contain itself, then by definition”, add “of “. After “On the other hand, if did not contain itself,” add “then by definition of “, and after “and hence”, add “by definition of “.
- Page 48: In the third to last sentence of Exercise 3.2.3, the period should be inside the parenthesis.
- Page 49+: change all occurrences of “range” to “codomain” (including in the index). Before Exercise 3.3.2, add the following paragraph: “Implicit in the above definition is the assumption that whenever one is given two sets and a property 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 as an ordered triple 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 , are said to be equal if and only if they have the same domain and codomain (i.e., and ), and for <I>all</I> .” 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 ” with “a given set “. Similarly, in Exercise 3.3.3 on page 55, replace “the empty function” with “the empty function into a given set “.
- Page 56: After Definition 3.4.1, replace “a challenge to the reader” with “an exercise to the reader”.
- Page 62: Replace Remark 3.5.5 with “One can show that the Cartesian product 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 is a set. (Hint: first use the axiom of replacement to show that for any , the set 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 -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 -tuple here to replace the construction in Exercise 3.5.1, or one can make a rather pedantic distinction between an ordered -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 of an ordered pair , 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 be an arbitrary set” after the first sentence, and let be a function from to rather than from to ; also should be an element of 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, would then have cardinality . But has equal cardinalit with (using as the bijection), hence , 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 is non-empty.
- Page 77: “negative times positive equals positive” should be “negative times positive equals negative”. Change “we call a
*negative integer*“, to “we call a*positive integer*and a*negative integer*“. - Page 89: In the first paragraph, insert “Note that when , the definition of provided by Definition 4.3.11 coincides with the reciprocal of 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 -closeness to include in the obvious fashion)”, and after Proposition 4.3.7, add “(extended to cover the 0-close case)”.
- Page 112: in the final display, take the square root around the integral (as in the penultimate display).
- Page 113: In the second paragraph of the proof of Proposition 5.4.8, add “Suppose that ” after the first sentence.
- Page 122: Before Lemma 5.6.6: “ root” should be roots”. In (e), add “Here ranges over the positive integers”, and after “decreasing”, add “(i.e., whenever )”. One can also replace by 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) .”
- Page 130: Before Corollary 6.1.17, “we see have” should be “we have”.
- Page 131: In Exercise 6.1.6, should be .
- Page 134: In the paragraph after Definition 6.2.6, add right parenthesis after “greatest lower bound of “.
- Page 138: In the second paragraph of Section 6.4, should be in math mode (three instances). After in the proof of Proposition 6.3.10, add “(here we use Exercise 6.1.3.)”.
- Page 140: In the first paragraph, should be in math mode.
- Page 143, penultimate paragraph: add right parenthesis after “ and 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 to . After Definition 6.6.1, add “More generally, we say that is a subsequence of if there exists a strictly increasing function such that for all .”.
- 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 when is defined on a larger set than . 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 should be replaced with its inverse (thus is defined by . 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, 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 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 as the difference of and . (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 (in particulaar is well-defined for any ).”.
- Page 178: In the proof of Lemma 7.5.2, after selecting , add “without loss of generality we may assume that “. (This is needed in order to take n^th roots later in the proof.) One can also replace and with and 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 ” should be “convergent for each “.
- 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 should instead be .
- Page 203: In Definition 8.5.8, “every non-empty subset of has a minimal element ” should be “every non-empty subset of has a minimal element “.
- Page 203: In Proposition 8.5.10, “Prove that is true” should be “Then is true”.
- Page 204: Before “Let us define a special class….”, add “Henceforth we fix a single such strict upper bound function “.
- Page 205: The assertion that is good requires more explanation. Replace “Thus this set is good, and must therefore be contained in ” with : “We now claim that is good. By the preceding discussion, it suffices to show that when . If this is clear since in this case. If instead , then for some good . Then the set is equal to (why? use the previous observation that every element of is an upper bound for for every good ), and the claim then follows since is good. By definition of , we conclude that the good set is contained in “.
- 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 ” should be “every element of “.
- Page 208: In Exercise 8.5.18, “Tthus” should be “Thus”. In Exercise 8.5.16, “total orderings of ” should be “total orderings of “.
- 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 is non-empty. In Exercise 9.1.9, delete the hypothesis that be a real number.
- Page 221: At the end of Remark 9.3.7, should be .
- Page 222: Replace the second sentence of proof of Proposition 9.3.14 by “Let be an arbitrary sequence of elements in that converges to .”
- 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 ” should be “drop the set “, and change to .
- Page 225: In Example 9.3.20, all occurrences of should be .
- Page 226: After Definition 9.4.1, add “We also extend these notions to functions that take values in a subset of , by identifying such functions (by abuse of notation) with the function that agrees everywhere with (so for all ) but where the codomain has been enlarged from to .
- 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 should be . In the sentence starting “Similarly, if …”, all occurrences of should be .
- 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 be a subset of , and let …”.
- Page 237: Add Exercise 9.6.2: If are bounded functions, show that , and are also bounded functions. If we furthermore assume that for all , is it true that 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 should be .
- 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, should be , and “also limit point” should be “also a limit point”.
- Page 257: In Definition 10.2.1, replace “Let …” with “Let be a subset of , and let …”. In Example 10.2.3, delete the final use of “local”. In Remark 10.2.5, should be .
- Page 259: In Exercise 10.2.4, delete the reference to Corollary 10.1.12.
- Page 260: In Exercise 10.3.5, should be .
- Page 261: In Lemma 10.4.1 and Theorem 10.4.2, add the hypotheses that , and that are limit points of respectively.
- Page 262. In the parenthetical ending in “$latex f^{-1} is a bijection”, a period should be added.
- Page 263: In Exercise 10.4.1(a), Proposition 9.8.3 can be replaced by Proposition 9.4.11.
- Page 264: In Proposition 10.5.2, the hypothesis that be differentiable on may be weakened to being continuous on and differentiable on , with only assumed to be non-zero on rather than . In the second paragraph of the proof “converges to ” should be “converges to “.
- 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 “ is nonempty and” before “the following property is true”, and delete the mention of the empty set in Example 11.1.3. In Lemma 11.1.4, replace “connected” by “either connected or empty”. (The reason for these changes is to be consistent with the notion of connectedness used in Analysis II and in other standard texts. -T.)
- In the start of Appendix A.1, “relations between them (addition, equality, differentiation, etc.)” should be “operations between them (addition, multiplication, differentiation, etc.) and relations between them (equality, inequality, etc.)”.
- Page 276: In the proof of Lemma 11.3.3, the final inequality should involve on the RHS rather than .
- Page 280: In Remark 11.4.2, add “We also observe from Theorem 11.4.1(h) and Remark 11.3.8 that if is Riemann integrable on a closed interval , then .
- Page 282: In Corollary 11.4.4, replace” ” by “, defined by “, and add at the end “(To prove the last part, observe that .)”
- Page 283: In the penultimate display, should be .
- Page 284: Exercise 11.4.2 should be moved to Section 11.5, since it uses Corollary 11.5.2.
- Page 288: In Exercise 11.5.1, (h) should be (g).
- Page 291: In the paragraph before Definition 11.8.1, remove the sentences after “defined as follows”. In Definition 11.8.1, add the hypothesis that be monotone increasing, and be an interval that is closed in the sense of Definition 9.1.15, and alter the definition of as follows. (i) If is empty, set . (ii) If is a point, set , with the convention that (resp. ) is when is the right (resp. left) endpoint of . (iii) If , set . (iv) If , , or , set equal to , , or respectively. After the definition, note that in the special case when is continuous, the definition of for simplifies to , and in this case one can extend the definition to functions that are continuous but not necessarily monotone increasing. In Example 11.8.2, restrict the domain of to , and delete the example of .
- Page 292: In Example 11.8.6, restrict the domain of to . In Lemma 11.8.4 and Definition 11.8.5, add the condition that be an interval that is closed, and 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 295: In the proof of Theorem 11.9.1, after the penultimate display , one can replace the rest of the proof of continuity of with “This implies that 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 ” with “all limit points of “. In the proof of Theorem 11.9.4, insert at the beginning “The claim is trivial when , so assume , so in particular all points of are limit points.”. When invoking Lemma 11.8.4, add “(noting from Proposition 10.1.10 that is continuous)”.
- Page 298: After the assertion , add “Note that , being differentiable, is continuous, so we may use the simplified formula for the -length as opposed to the more complicated one in Definition 11.8.1.”
- Page 299: In Exercise 11.9.1, should lie in rather than . In Exercise 11.9.3, should lie in rather than . 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, may be improved to ; 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 to indicate that a mathematical object is being identified with a mathematical object .”
- Page 334: In the last paragraph of the proof of Theorem B.1.4, “the number has only one decimal representation” should be “the number has only one decimal representation”.

Note that the first edition paperback page numbers differ from the second (or third) edition hardback page numbers, which should be born in mind when applying the second edition errata to the first edition. (The section, theorem and exercise numbering, however, is mostly unchanged.)

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

## 1,022 comments

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3 July, 2021 at 11:22 am

Darshan PillayDear Professor Tao,

I am currently using the 3rd corrected edition of Analysis 1 and I am having some trouble understanding Definition 9.3.6 (Convergence of functions at a point) and also Definition 9.3.3 (Local epsilon-closeness). In particular when we set X to be the empty set then X has no adherent points and I am not sure how to make sense of Definition of 9.3.3 and Definition 9.3.6 in this case. In the Definition 9.3.3 do we impose that X is non-empty and in Definition 9.3.6 do we leave the limit undefined when X is empty?

Kind Regards,

Darshan Pillay

3 July, 2021 at 11:26 am

Darshan Pillayin Definition 9.3.6 do we also require that E be non-empty?

6 July, 2021 at 12:34 am

Terence TaoThe non-emptiness of or is a consequence of the hypotheses of the definition (since, as you say, there would otherwise not be any adherent points), but does not need to be explicitly stated; there is no way in which Definition 9.3.6 could be applied to an empty set since one could not produce the adherent point that is part of the concept being defined.

6 July, 2021 at 6:53 am

yalikesDear Professor Tao,

I am reading the 3rd edition of Analysis 1, and I have a question about the definition of -length in Riemann-Stieltjes integral(in the Errata to the third edition, definition 11.8.1),

Do we need function to be bounded or domain of X to be closed interval?

you see, if and

does not define.

This means for some , -length didn’t define for some bounded interval I. length of interval always have a definition, but

doesn’t. Is this intended?

I am confusing about this.

(English is not my native language, sorry).

[Fair enough, I have now required to be closed. -T]13 July, 2021 at 10:57 am

William DengIf is now required to be closed, then would the domain in Examples and , or the domain in Examples and be considered closed? They wouldn’t from the perspective of Definition , but would be relatively closed with respect to (as in Section of Analysis II).

[Here we are using the notion of closed set from Definition 9.1.15; I’ve updated the errata accordingly. -T]11 July, 2021 at 12:17 am

AnonymousDear prof tao.

in Proposition 5.5.12, the book says “Since x^2 >2, we can choose 0<e2 , and thus (x-e)^2 >2”

how can we choose an e??

13 July, 2021 at 12:50 pm

William DengIn Exercise , after the map is defined, it says “that sends every element of to its order ideal” when it should say “that sends every element of to its order ideal”. In Exercise , there is a part that read “Show that the maximal elements of are precisely the total orderings of ” when it should instead be “the total orderings of “.

14 July, 2021 at 12:07 pm

JTerry, this page used to load all the comments at once and one can easily search among them conveniently. Now it loads only a few. Do you know a way to search the comments conveniently (e.g. find all the comments by “Terence Tao”)?

16 July, 2021 at 9:35 am

William DengIn Definition , don’t we technically also need to know the nature of the equality relation which has been defined on the objects of in order to check anti-symmetry for instance? So instead of viewing a poset as just a pair , we should view it as , where is the equality relation which has been defined on the elements of ?

[All first-class mathematical objects are understood to have an equality relation assigned to them. -T]20 July, 2021 at 10:20 am

William DengIn Zorn’s lemma (Lemma ), is the assumption that really necessary? I believe the other hypothesis that every chain in has an upper bound ( ) is sufficient to ensure that .