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Basic Analysis: Introduction to Real Analysis: Errata

This page lists errors in the various editions. Nonmathematical typos, misspellings, and grammar or style problems are not listed here. Also not listed are things that were correct but simply deserved improvement. Of course, older editions may suffer from recently discovered errata as well.

February 29th 2016 edition, Version 4.0 (edition 4, 0th update):

No known errata.

December 16th 2014 edition:

  1. On page 22, axioms (A2), (A3), (M2), (M3) should not begin with an "If". Thanks to Jim Brandt.
  2. On page 23, top of page, it should also say that (A2), (A3), and (A4), have been used along with (A5).
  3. On page 27, end of proof of Theorem 1.2.4 parth (ii), there is a typo, a < is written when $\leq$ was intended. The penultimate sentence of the proof should be "If $y \leq 0$, then note that $0 \leq -y \lt -x$ and find a rational $q$ such that $-y \lt q \lt -x$."
  4. On page 34, Exercise 1.3.7 requires arithmetic with extended reals, which we explicitly said we'll avoid. In the next edition, there will be the extra assumption of "bounded", which is what was really intended for the exercise.
  5. On page 35, tangent is 1-1 and onto on $(-\pi/2,\pi/2)$, not $(-\pi,\pi)$.
  6. On page 80, second line from the bottom, it is $r^k$ not $r^n$ all the way on the right hand side.
  7. On page 84, in the proof of the alternating series test it says "Similarly, $(x_{2k}-x_{2k-1}) \geq 0$" that should say "Similarly, $(x_{2k}-x_{2k+1}) \geq 0$". Thanks to Trevor Mannella for pointing this out.
  8. On page 88, a minor typo, $m \geq 2K$ implies $m-K+1 > K$, not $K+1$, but that is all that is needed.
  9. On page 102, Exercise 3.1.9, an extra assumption is necessary, for example $g(c_2)=L$ (i.e. continuity of $g$ at $c_2$). Thanks to Gregory Beauregard for pointing this out!
  10. Similarly on page 122, Proposition 3.5.8. If $b \in B$, then we must assume that $g(b) = c$.
  11. On page 107, Example 3.2.12, in the proof that the Thomae function is continuous on the irrationals we take a sequence of rationals, though we should really take an arbitrary sequence. Still only finitely many of the numbers which happen to be rational have denominator larger than K. Thanks to Jim Brandt.
  12. On page 114, Exercise 3.3.6, the degree of the polynomial had better be positive.
  13. On page 122, in end of example 3.5.7, we mean that $\frac{x}{2} > N$ not $\frac{x}{2} > M$.
  14. On page 125, in proof of corollary 3.6.3, $f(x_1) \leq a$ and $f(x_2) \geq b$, that is, the inequalities should be nonstrict.
  15. On page 125, corollary 3.6.3 should really assume that $f$ is not constant otherwise $f(I)$ is a single point, and we defined intervals in such a way as to exclude single points. (Thanks to Andreas Giannopoulos)
  16. On page 141, In definition 4.3.1, the first expression for the Taylor polynomial mistakenly starts at $k=1$ instead of $k=0$. The second expression is correct.
  17. On page 143, Exercise 4.3.5. The divisor in the expression is $(x-x_0)^n$ not $x^n$. The exercise is still doable of course but it is only the intended thing when $x_0 = 0$.
  18. On page 153, in example 5.1.14 there are a couple of typos. First, when $x_j$ is $\frac{jb}{n}$ not $\frac{ib}{n}$ (also similarly a few lines below). Second when computing $M_j$, the formula should be $\frac{1}{1+x_{j-1}}$. And finally $\Delta x_j$ is pulled out of the sum, but while in some sense OK as $\Delta x_j$ does not depend on $j$, it should only be pulled out after we substitute for it with $\frac{b}{n}$.
  19. On page 155, Exercise 5.1.9 is just an exact duplicate of 5.1.8. It will be replaced in the new version by a different exercise.
  20. On page 164 there the $c_i$ is really in $(x_{i-1},x_i)$, which is the conclusion of MVT, and we do need $c_i$ not to be $a$ nor $b$.
  21. On page 180, Exercise 5.4.4. The formula for the geometric sum is missing a $t^{n+1}$ on the second term on the right hand side, that is, it should be $1-t+t^2-\cdots+{(-1)}^n t^n = \frac{1}{1+t} - \frac{{(-1)}^{n+1}t^{n+1}}{1+t}$.
  22. Minor typo on page 184 about middle of the page. It says $\cdots < \epsilon/2 + \epsilon/2 < \epsilon$, but clearly $\cdots < \epsilon/2 + \epsilon/2 = \epsilon$.
  23. On page 194, Exercise 6.1.9, the $f(0)=0$ should be $f_n(0)=0$. (Thanks to Chase Meadors)
  24. On page 200, Exercise 6.2.12, the functions are from $[0,1]$, the domain was missing.
  25. On page 211, in the proof of the triangle inequality for $C([a,b],\mathbb{R})$ the computation starts with $d(f,h)$ when it should start with $d(f,g)$ (the definition is unrolled correctly however).
  26. Exercise 7.1.5: increasing is not needed, also the function is defined on $[0,\infty)$ not $[0,\infty]$.
  27. On page 228, $B(x_2,\delta) \in U_{\lambda_2}$ should of course be $B(x_2,\delta) \subset U_{\lambda_2}$.
  28. On page 230, in proof of Proposition 7.5.2, the metrics should be $d_X$ and $d_Y$ and not just $d$.
  29. On page 234, in Theorem 7.6.2, we say "f has a fixed point" but it should say "f has a unique fixed point". We actually prove uniqueness, and we use uniqueness later. (Thanks to Arthur Busch)
  30. On page 234, bottom of page in the uniqueness proof, the second equality should be an inequality, that is $d(f(x),f(y) \leq k d(x,y)$.
  31. On page 235, just above Picard's theorem, when we say $f'(x_0) = y_0$ we should say $f(x_0) = y_0$. (Thanks to Arthur Busch)
  32. In the proof of Picard we say $f([-h,h]) \subset [x_0 -\alpha,x_0+\alpha]$ when that should be $f([-h,h]) \subset [y_0 -\alpha,y_0+\alpha]$. (Thanks to Arthur Busch)
  33. On page 236, about middle of the page the estimate $|t|Ld(f,g)$ should of course be $|x|Ld(f,g)$.

October 20th 2014 edition:

  1. In exercise 5.3.6, we need to assume that $F'$ and $G'$ are integrable. A simple fix is to just assume that F and G are continuously differentiable. Thanks to Kristopher Lee and Hannah Lund for pointing this out.

December 18th 2013 edition:

  1. In exercise 2.6.11, the definition of $a_n$ is missing something in part c) making the series not Cesaro-summable. You should let $a_n := k$ if $n=k^3$, $a_n := -k$ if $n=k^3+1$, and $a_n:=0$ otherwise. The $-k$ part was missing. Thanks to Kristopher Lee and Baoyue Bi for noticing this.

October 7th 2013 edition:

  1. In exercise 7.1.8, the Hausdorff metric is not a metric on bounded sets but only a pseudometric. That is, $d_H(A,B) = 0$ does not mean that $A=B$ (It would work for compact sets, but we don't know those yet in 7.1). Thanks to Kenji Kozai for spotting this.

May 29th 2013 edition:

  1. In exercise 5.2.16, the second inequality (for lower darboux integral) should be reversed. It should be $\underline{\int_a^b}(f+g) \geq \underline{\int_a^b}f+\underline{\int_a^b}g$. (Thanks to Sonmez Sahutoglu)
  2. On page 172, in the proof of Proposition 5.4.2, when proving property (iii) we say "$E(x_0) \lt \epsilon$ for all $x \leq x_0$". Of course, that should be "$E(x) \lt \epsilon$ for all $x \lt x_0$" (the rare appearance of two, although minor, typos in one sentence).
  3. On page 145, in corollary 4.4.3, it says "there exists a unique positive number" even though we allow $x \geq 0$, so we have to allow the root to be 0.

December 16th 2012 edition:

  1. In exercise 5.3.5, page 137, it should be OK to assume "continuously differentiable". The exercise is possible, but it is harder than intended. (Thanks to Sonmez Sahutoglu)
  2. On top of page 148, the second equality is an inequality. It is not difficult to see, but it requires a proof. I will be adding a relevant exercise. (Thanks to Sonmez Sahutoglu)
  3. Page 150, exercises 6.2.10 and 6.2.11. The assumption of "continuous" on $f_n$ is missing in both of these exercises. They're not true without it. (Thanks to Sonmez Sahutoglu)
  4. Page 174, exercise 7.3.10: part b) is not correct. The metric given is not the right metric for this problem. The problem will be replaced with a different and more comprehensive one. (Thanks to Sonmez Sahutoglu)

October 1st 2012 edition:

  1. In exercise 2.5.9 on page 78, the two series should be absolutely convergent, or part a) is not true. (Thanks to Sonmez Sahutoglu)
  2. On page 73, second equation from the top we say "triangle inequality," but we write an equality. Of course that should be an inequality. (Thanks to Yi Zhang)
  3. On page 76, 6th line from the bottom, an $r^n$ should be $r^k$, we get it right in the line just below. (Thanks to Yi Zhang)
  4. In Theorems 3.3.7 and 3.3.8, c is really in the open interval (a,b), though that's obvious. It's correct as stated, but stupid. So change to $c \in (a,b)$. Similarly in the application on page 95. (Thanks to Yi Zhang)
  5. In remark 5.1.5, we meant integrals from a to b, not 0 to 1.

April 8th 2012 edition:

No known errata.

December 25th 2011 edition:

No known errata.

December 15th 2011 edition:

  1. Yaikes! The triangle inequality on page 157 has a typo. Of course it is an inequality, otherwise it would be called the triangle equality. So it should say $d(x,z) \leq d(x,y) + d(y,z)$. Thanks to students from my class.
  2. When renaming t to x in section 7.6, in the definition of T(f) on page 185, I forgot to change t to x. It says T(f)(t) when it should say T(f)(x).

November 18th 2011 edition:

  1. On page 164 in the definition of an open set. It should say "... if for every $x \in V$, there exists ..." Instead it says for every $x \in X$ which is of course wrong. (Thanks to Steve Hoerning)
  2. Page 183, in the contraction mapping principle, of course X should be nonempty. I don't feel too bad about making this mistake as Rudin does it also, even in the third edition.
  3. Page 181, in Theorem 7.5.4, f is bounded not compact, it is f(X) that is compact.
  4. Exercise 7.1.8 works only for nonempty bounded sets. The definition works if you use extended reals, but you won't get a metric space.
  5. Page 166, in the proof of proposition 7.2.8 in the middle of the estimate there is $-\delta$ when there should be $+\delta$.
  6. In the proof of proposition 7.2.13 we actually need the opposite direction of Proposition 7.2.11. So 7.2.11 needs to be made into an if and only if. The other part of the if and only if simply refers to exercise 7.2.12. To do this we also need to change the assumption on the intersection to $U_1 \cap U_2 \cap S \not= \emptyset$. Consequently in the proof we show that $U_1 \cap S$ and $U_2 \cap S$ have a point in common (which we actually do).
  7. In the proof of proposition 7.2.13 we are working with [x,y] in the end not with S, that's the whole point of defining x and y.
  8. On page 177 in the middle when defining $n_{j+1}$ it should be in the ball of radius 1/(j+1) not 1/j. Or alternatively we should start with $n_{j-1}$ defined and define $n_j$ to be in the ball of radius 1/j.
  9. In proof of proposition 7.3.7, on the last line we say $n \geq N$ when of course that is $n \geq M$ (there is no N in the whole proof).
  10. In exercise 7.4.1, A is of course finite subset of X. (Thanks to Jeremy Sutter)
  11. In the proof of Theorem 7.4.6 (sequentially compact is equivalent to compact) we define a sequence starting at $x_0$ and $\lambda_0$, but then several times we forget it later and write down the elements starting at $x_1$ and $\lambda_1$. Of course we should consistently start with 0 (or even better with 1).
  12. In exercise 7.6.5, the "best" k is of course the smallest one, not the largest one (there is no largest one of course).
  13. Definition 1.1.1 on page 14 was not well stated. An ordered set is the set together with the relation. (Thanks to Paul Vojta)
  14. On page 106, in the statement of the chain rule, $c \in I_1$ not $c \in I_2$. (Thanks to Paul Vojta)
  15. On page 107, in exercises 4.1.3 and 4.1.8 it was not specified what is n. So specify that it is an integer, in which case also specify that x and f(x) should not be 0. (Thanks to Paul Vojta)
  16. On page 112, in the proof of Proposition 4.2.6, we assume that $x < y$ but then we use that $x - y > 0$ when we should use $x - y < 0$. (Thanks to Paul Vojta)
  17. On page 134, the second displayed equation is only true when $c < x$, otherwise the opposite inequalities hold. (Thanks to Paul Vojta)

October 16th 2011 edition:

  1. On page 96, in the proof of Proposition 3.3.10, near the end there is a missing absolute value sign, or at least a negative sign. That is, after the "or in other words", there should be $-(b_{d-1}M^{d-1}+\cdots+b_1M+b_0) < M^d$. Just above, there should be absolute value signs, as in $| ... | < 1$. Similarly in the following paragraph the typo was propagated. So for the $K$ it should say $b_{d-1} (-K)^{d-1} + \cdots + b_1 (-K) + b_0 < K^d$
  2. On page 93, in proof of Theorem 3.3.2. When we say "there exist convergent subsequences $\{ x_{n_i} \}$ and $\{ y_{n_i} \}$". The indices should be different. Now it is possible to pick one $n_i$ for both, so it is not really wrong, but there is no need to do this. Simply change $y_{n_i}$ to $y_{m_i}$.
  3. Exercise 3.3.7, says image of a closed and bounded interval is a closed and bounded interval. I suppose it is OK, but some would not consider $[c,c] = \{c\}$ an interval. So best to suppose that f is nonconstant.
  4. On page 94, about 3/4 down. d is the limit of $\{ b_n \}$ which is decreasing, so it's the infimum, not the supremum (Thanks to Daniel Pimentel-Alarcón).
  5. Page 106, the quotient rule is of course missing a minus sign. (Thanks to Jeremy Sutter)
  6. On page 59, line 5, that should be $\{ x_n : n \geq n_k+1 \}$. (Thanks to Eliya Gwetta)
  7. On page 120, in the proof of 5.1.7, the $\tilde{m}_j$ definition is not quite right, a tilde is missing. It should be $\tilde{m}_j := \inf \{ f(x) : \tilde{x}_{j-1} \leq x \leq \tilde{x}_j \}$.
  8. On page 128, in the proof of Lemma 5.2.6, we wish to take an $n$ such that $\frac{b-a}{n} < \delta$, the (b-a) was missing.
  9. On page 122, when computing the difference of upper and lower integral it is $(1+\epsilon)-(1-\epsilon)$ of course.
  10. On page 128, in the definition of "finitely many discontinuities". Of course we mean that f is continuous at all points of A, and not that the restriction of f to A is continuous.
  11. On page 143, there were absolute value signs missing in the estimate towards the end of proof of 6.1.13. That is there should be absolute value signs around $f(x)-f_k(x)$.
  12. On page 145 and 147, it says that $n > \frac{1}{x}$ implies $x < \frac{1}{n}$, while that should of course be $x > \frac{1}{n}$, which is why $f_n(x)=0$.
  13. On page 149, Exercise 6.2.3, of course the function f should be Riemann integrable to be able to take the integral. Similarly for 6.2.8 and 6.2.9 on page 150, the functions $f_n$ should be Riemann integrable.
  14. On page 136, the definition of the erf function is wrong. A minus sign and a square root is missing. It is simply an example, and not used elsewhere in the book.
  15. On page 75, we mention that $\sum \frac{1}{n^2}$ converges to $\frac{\pi^2}{2}$ when in fact it converges to $\frac{\pi^2}{6}$ (Thanks to Daniel Pimentel-Alarcón for spotting this).
  16. In the examples on pages 154 and 155, the h guaranteed by the proof of the theorem was computed for an older version of the proof. For exponential, $h=\frac{1}{2}$ will work and for $y'=y^2$, $h=1-\frac{\sqrt{3}}{2}$ will work.
  17. On page 143, the strict inequality in the displayed equation in the middle of the page should of course be a nonstrict inequality as we took a limit.
  18. In example 5.3.2, the computation has a typo, while the final answer is correct.
  19. In example 5.1.12, at the end of the example, there is $\int_0^1 f$ when there should of course be $\int_0^2 f$ (Thanks to Jeremy Sutter).
  20. On page 135, the difference quotient is less than or equal to epsilon, not strictly less than.
  21. In exercise 6.1.3, A is of course a subset of S, not just a subset of the real numbers, otherwise the restrictions don't make sense.
  22. In exercise 5.2.10, f should be bounded as well, otherwise it obviously does not make any sense (Thanks to students in my class)

April 26th 2011 edition:

  1. Proposition 1.2.7, page 28: The set A should be bounded and nonempty, it is true without this hypothesis, but we just made a big deal about not doing arithmetic with ${\mathbb{R}}^*$.
  2. Proposition 1.2.8, page 29: A, B must be nonempty sets!
  3. In the proof on top of page 29, all inequalities should be nonstrict.
  4. Exercise 2.3.6, page 63: The sequences should be bounded. The exercise works with unbounded, but we have not defined limsup and liminf for unbounded sequences.
  5. Example 2.5.18, page 77 is misleading. It is not true that the series converges because the terms go to zero, but by the way we proved that the terms go to zero (using the ratio test). Better replace the justification by the actual ratio $ \lim_{n\to\infty} \frac{2^{n+1}/(n+1)!}{2^n/n!} = \lim_{n\to\infty} \frac{2}{n+1} = 0 . $
  6. Exercises 1.2.9 and 1.2.10 should say that A and B are nonempty (Found by Paul Vojta)
  7. Exercise 2.4.3 has a typo. The rational numbers need to be dense in F. Here is a rewritten version that is also a lot more explicit.
    Exercise 2.4.3: Suppose that F is an ordered field that contains the rational numbers $\mathbb{Q}$, such that $\mathbb{Q}$ is dense, that is: whenever $x,y \in F$ are such that $x < y$, then there exists a $q \in \mathbb{Q}$ such that $x < q < y$. Say a sequence $\{ x_n \}_{n=1}^\infty$ of rational numbers is Cauchy if given any $\epsilon \in \mathbb{Q}$ with $\epsilon > 0$, there exists an M such that for all $n,k \geq M$ we have $|x_n-x_k| < \epsilon$. Suppose that any Cauchy sequence of rational numbers has a limit in F. Prove that F has the least-upper-bound property.
  8. In exercise 2.5.6 it should say "converges absolutely" in a) it just says "converges". Of course that's not wrong (it's weaker), but it could be misleading, and the way to prove convergence here is to prove absolute convergence.
  9. On page 85, second line it says "thus c is a cluster point of A", of course that is supposed to be "cluster point of S" as that's what we are trying to prove.
  10. On page 110, end of proof of Theorem 4.2.3. Is says "Hence the relative minimum is 0 and the relative maximum is 0" Those are of course the "absolute" min and max.
  11. On page 112: In proposition 4.2.8, the second interval is (c,b) not (c,d). There is no d in the proposition.

February 28th 2011 edition:

  1. On top of page 102, the $=\frac{1}{2}|x-y|$ should of course be $\leq\frac{1}{2}|x-y|$.

December 26th 2010 edition:

  1. Exercise 1.1.7 has a extra subset sign. $A \subset \subset \mathbb{N}$ should be $A \subset \mathbb{N}$.
  2. On page 12, the sentence about induction hypothesis was missing the word "in", that is "assumption in" rather than just "assumption". It is "P(n) is true" that is the hypothesis.

November 1st 2010 edition:

  1. On page 11, "So let us assume that $x \in A \cap (B \cup C)$" should be "So let us assume that $x \in A \setminus (B \cup C)$". Thanks to Dan Stoneham.
  2. On page 80, first line of subsection 3.1.2 c is of course a cluster point of S, not A.

October 3rd 2010 edition:

  1. In exercise 1.3.4, the functions f and g are of course bounded as in proposition 1.3.7. (Glen Pugh)
  2. Proof of proposition 2.2.5 part iii was off if $y=0$. We should use $\frac{\epsilon}{2(|y| + 1)}$ instead of $\frac{\epsilon}{2|y|}$. (Glen Pugh)
  3. The function defined in the beginning of Example 6.2.3, was not the one on Figure 6.3 nor the rest of the example (though it could also be used to show the same idea).
  4. In example 4.2.10, the argument showing that f' is not continuous at zero was insufficient. That the limit does not exist is not as obvious as it seemed.

September 6th 2010 edition:

The following errata were found by Glen Pugh.

  1. In example 0.3.14, $f^{-1}(\{0\}) = \pi \mathbb{Z}$ not $\mathbb{Z}$.
  2. In proposition 0.3.16, C and D are of course subsets of A, not B.
  3. In exercise 0.3.4 part b), the explanatory sentence has a typo, it should be that "$f( C \cap D)$ is a proper subset of $f(C) \cap f(D)$"

August 12th 2010 edition:

  1. On bottom of page 11, the counter-example for swapping intersection and union doesn't work due to a typo. The set should be defined by $\{ k \in \mathbb{N} \mid mk < n \}$ not by $\{ k \in \mathbb{N} \mid k < nm \}$. Thanks to Glen Pugh for spotting this.

July 15th 2010 edition:

  1. Exercise 5.2.11 mentioned as a side note that the Thomae function is "everywhere discontinuous," which is a typo of course. It is only discontinuous on a dense set. (Thanks to an anonymous reader)

June 23rd 2010 edition:

No known errata.

April 8th 2010 edition:

All the below errata were found thanks to Jana Maříková.

December 23rd 2009 edition:

No known errata.

December 11th 2009 edition:

Proposition 5.2.4 had a typo in it. There was one too many alphas in there.

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