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I preserve numbering as much as possible. So for example exercises are only added with new numbers, so that old exercises are not renumbered, etc... I try to preserve pagination as well though adding a page in the middle is sometimes unavoidable.

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

The main point of this revision is to improve places where it seemed a bit too terse and dry. So I added a number of small examples throughout, especially in the beginning of the book, certain explanations and proofs were expanded or rewritten, and quite a few new figures were added, to illustrate expamples or proofs. It also adds a number of new exercises.

Mostly due to the added content page numbers and
some figure numbers have changed,
in section 2.6 a numbered example 2.6.9 was added, which changed
numbers on the two propositions and an example following.
**Existing exercise numbers are the same, except that**
some exercises were added and,
**exercises 4.4.2, 5.1.9, and 7.6.3 were replaced**.
That is, the new edition is essentially backward compatible as usual.

Here is a detailed list of the larger changes:

- Add a couple of examples to section 1.1 on different types of ordered sets so that students are not lulled into the sense that only numbers can be ordered. Also mention already in this section that the rationals are an example of an ordered field.
- Also add a couple more quick concrete examples of subsets of the rationals and their upper/lower bounds or lack thereof after Definition 1.1.2.
- Add Exercise 1.1.10 on computing supremum in an infinite dictonary.
- Very slight simplification in the proof of Theorem 1.2.4 part (ii) as $x < \frac{m}{n}$ actually is proved earlier and so disentangle it from the proof of $\frac{m}{n} < y$
- In Proposition 1.2.6, put the right sort of boundedness of $A$ into the statements as needed. Then make the note about $x+A$ and $xA$ being nonempty and bounded match this (and be more explicit). This really fixes a logical gap in the exposition.
- Add Exercise 1.2.12 to prove Proposition 1.2.8, and exercise 1.2.13 to prove Bernoulli's inequality.
- Compute the inf and sup of the example function after Definition 1.3.6 and the definition of inf and sup of a function.
- In the proof of Proposition 1.5.1, add a displayed inequaity to make the derivation of the inequality for $n+1$ more explicit.
- In Exercise 1.3.7 we add a hypothesis of "bounded" to avoid use of arithmetic on extended reals, which we have not defined and said we would avoid just a few pages earlier.
- Add Exercise 1.4.8 to show cardinalities of [0,1] and (0,1) are equal by constructing injections only, also add Exercise 1.4.9 to prove there are only countably many algebraic numbers.
- Add a figure (actually two figures) illustrating the convergence of a sequence, that is the role played by the limit, the epsilon and the M.
- After definition of monotone sequences, add a small paragraph with a couple of quick words of which sequences we have seen are monotone and which are not monotone sequnces. Plus add a sample "graph" figure of an increasing sequence.
- Make Proposition 2.1.15 into a TFAE type of statement with 3 statements, to give students practice with these. Thanks to Sonmez Sahutoglu for the suggestion.
- Add a quick example of a non-monotone sequence whose 3-tail is monotone to illustrate use of Proposition 2.1.15
- Add exercises 2.1.19, 2.1.20, 2.1.21, 2.1.22
- Slightly simplify the proof of the division claim in the proof of Proposition 2.2.5.
- After proposition 2.2.7, add a small example of the use of the continuity propositions.
- In the recursive sequences subsection mention that the example is Newton's method (and add Wikipedia link to Newton) and mention that it can work to compute square roots more generally, with the explcit statements left as an exercise.
- Add exercise 2.2.11, 2.2.12
- Add figure to illustrate proof of Lemma 2.2.12, and expand on possible counterexamples when L=1.
- Move the note about boundedness and monotonicity of $a_n$ and $b_n$ from definition 2.3.1, and make it into the first item in Proposition 2.3.2.
- In section 2.4, add an extra remark about
- Add exercises 2.4.6, 2.4.7, 2.4.8
- In section 2.5 among other minor improvements, more consistently use $n$ for index of the terms and $k$ as index of the partial sums to avoid switching the meaning of $n$ too often.
- Add exercises 2.5.12, 2.5.13
- Add Example 2.6.9, this moves the numbers of the following two propositions and the example.
- Add exercises 3.1.13, 3.1.14
- Add a couple of inline examples in 3.1.
- In proof of proposition 3.2.4, do the computation in reverse as that is the way it should be understood. Same in 3.2.3.
- Add Example 3.2.13, removable discontinuity, and contrast this to the step function.
- Add exercises 3.2.14, 3.2.15, 3.2.16
- Add exercises 3.3.11, 3.3.12, 3.3.13
- Add figure for proof of Lemma 3.3.7 (bisection method).
- Add figure for definition of Lipschitz in section 3.4.
- Add exercises 3.4.10, 3.4.11, 3.4.12, 3.4.13, 3.4.14
- Fixed the error in Propositon 3.5.8.
- Add figure 3.6 to illustrate corollary 3.6.3, but it also illustrates monotone functions and continuity in general.
- Add more detail throughout 3.6.
- Add exercise 4.1.12
- Add two figures in section 4.2, first one illustrating the proof that a relative extremum is a critical point and the second illustrating the function with a discontinuous derivative.
- Add very short example in 4.2 showing that differentiability is necessary for Rolle's theorem.
- In Rolle's theorem, don't assume $f(a)=f(b)=0$, simply assume $f(a)=f(b)$. I am not sure why I did that originally. Since I was at it I also rewrote the proof Rolle's to be a little clearer.
- Add exercises 4.2.11, 4.2.12.
- Add exercises 4.3.7, 4.3.8.
- In 4.4 add a figure to the proof of Lemma 4.4.1, also change Lemma 4.4.1 to be the pointwise result.
**Replace exercise 4.4.2**as Lemma 4.4.1 is now the pointwise result.- Add exercise 4.4.8
- Improve the basic Riemann sum figure.
**Replace exercise 5.1.9**(which was a duplicate of 5.1.8) with a new one.- Add exercises 5.1.12, 5.1.13, 5.1.14
- After Proposition 5.2.4, add a note about only getting inequalities for the upper and lower integrals for sums, and refer to the exercise.
- Add a couple of quick examples about discontinuity and differentiability after second form of FTC.
- Add exercise 5.3.12
- After definition 6.1.9 note that S is important and also mention some other notations and names for the uniform norm.
- Streamline the proofs in Example 6.1.8 and Proposition 6.1.13.
- Add Example 6.2.7, pointwise convergence of bounded functions does not imply even bounded.
- At end of 6.2 mention that derivatives need stronger convergence and point to the exercises.
- Add exercises 6.2.13 and 6.2.14 about joint limits.
- Exercise 7.1.5, remove the increasing hypothesis it is not necessary.
- In Proposition 7.2.14 add nonempty as hypothesis. Better that than to rely on a technicality of our definition of connected.
- Include the reverse direction in proposition 7.3.10, and refer to exercise 7.3.1 in the proof.
- In propositions 7.3.7 and 7.4.4, stop using superscripts for sequences in ${\mathbb{R}}^n$. Seems to confuse students more, so use a double subscript. Perhaps that works better, they are more used to it.
- Add Example 7.4.12 (discrete metric counterexample), and a few add notes on the results throughout 7.4
- Add Lipschitz functions as Example 7.5.11.
- Split 7.6 into subsections, also explicitly state we are going to work with $C([a,b],{\mathbb{R}}$ before delving into the proof.
- Add some simple examples in 7.6.
- Rework the the proof of Picards a little bit, that is make $Y$ be the
more natural space of all functions mapping into $J$ and add more details to
the proof. This simplifies the proof and removes the need for Exercise 7.6.3,
so
**replace 7.6.3**with another easy exercise. - In exercise 7.6.7 mention explicitly the iteration scheme is from the fixed point theorem, which is in fact the same iteration scheme as in chapter 6.
- Add exercises 7.6.8, 7.6.9, and 7.6.10, and reword some of the other exercises to be clearer.
- There were several labeled equations in chapter 7 that were never referred to, removed those labels.
- Some very minor improvements in style and exposition throughout.
- Typos fixed.
- Fix the errata from the last revision.

You can download the LaTeX source

**December 16th 2014 edition:**

- Some very minor improvements in style and exposition.
- Add note to exercise 5.3.6 to compare to 4.2.8.
- Fix the erratum from the last revision.

You can download the LaTeX source or PDF for this old version if you want (but I recommend just using the current version).

**October 20th 2014 edition:**

- Many minor improvements in grammar, style, and exposition.
- Fix the erratum from the last revision, that is, Exercise 2.6.11 part c) was unsolvable due to a missing piece of the definition of $a_n$.

You can download the LaTeX source or PDF for this old version if you want (but I recommend just using the current version).

**December 18th 2013 edition:**

- Add uniform continuity to section 7.5 and add Theorem 7.5.10 to prove that function continuous on a compact set is uniformly continuous.
- Fix exercise 7.1.8 to ask only for a pseudometric space, and add a part asking for an example of degeneracy. Then add exercise 7.4.15 to prove that the set of compact sets is a metric space. Thanks to Kenji Kozai for the suggestion.
- Many minor improvements in grammar, style, and exposition.

You can download the LaTeX source or PDF for this old version if you want (but I recommend just using the current version).

**October 7th 2013 edition:**

- Fix the errata from the last revision.
- Fix some grammer/english issues.
- Add some minor clarifications.
- Actually define the words "Cauchy product" in the new series section, and put it in the index.
- Example 3.3.9 is the "bisection method", so give this name and put it in the index.
- In exercise 5.4.4, in the "Finally" comment, the answer was included leaving the reader possibly confused as to what they are to do.
- In chapter 6, there were quite a few places where a sequence of functions was simply referred to as $f_n$ rather than $\{ f_n \}$, so fix that.
- Go on an anti-"that" offensive as it is overused and just contributes to "wordiness" in many places. (I feel it is still overused but it is now somewhat better)

**May 29th 2013 edition:**

Many of the additions in this version were inspired by feedback from Sonmez Sahutoglu, and also by browsing the additions in the University of Pittsburgh version, so they deserve much credit in making me write these new sections.

Numbering changes as little as possible. Obviously page numbers are changed
and new sections and propositions were added, but none of the new ones
change numbering of previous sections, propositions, examples, or exercises.
Other than that the only relevant numbering changes are
that **exercises 1.4.3 and 7.3.10 were replaced**.

**Add optional section 1.5**on decimal expansion. This makes the hard exercise 1.4.3 obsolete, so replace with another exercise.**Add optional section 2.6**on more topics on series.**Add optional section 3.5**on limits at infinity.**Add optional section 3.6**on monotone functions and continuity.**Add optional section 4.4**on inverse function theorem.**Add optional section 5.4**on the logarithm and the exponential.**Add optional section 5.5**on improper integrals.- Add Proposition 5.1.13 and another example 5.1.14 to show integrability directly.
- Add a very short subsection 2.3.4 about infinite limits of sequences and add exercises 2.3.13 and 2.3.14 to go with it.
- Add exercises 1.3.7, 1.4.6, 1.4.7, 2.5.11, 3.3.10, 4.3.5, 4.3.6, 5.2.16, 6.3.6.
- In 3.1.4 it is natural to define one sided limits (definition 3.1.15), do so and add proposition 3.1.16 and exercise 3.1.12
- In exercise 7.3.5, onto is not needed, and might confuse the student, so remove the hypothesis.
- Mark the dependence on x
_{0}on the Taylor polynomial and remainder. - Add links to A Gentle Introduction to the Art of Mathematics, and Book of Proof for the proof based courses in the book intro. Being a free book we should recommend other good free books.
- In references to famous mathematicians link the Wikipedia article. Also we completely lacked such a footnote for Cantor.
- Removed the correspondence of sections with [BS]. I don't think it's very useful anymore (it was originally for my students when the book was really just a set of lecture notes), it is getting more complicated anyway.
- Fix the errata from the last revision (In particular exercise 7.3.10 was replaced).
- Minor English and style fixes.

**December 16th 2012 edition:**

- Fix the errata from the last revision.
- Footnote counter is reset per page as is usual, so that we don't go into weird footnote symbols.
- In Taylor's theorem proof make it explicit that M and hence c depends
on x and x
_{0}. (Thanks to Sonmez Sahutoglu for the suggestion) - Add exercises 3.3.9, 4.3.4, 5.3.10, 5.3.11

**October 1st 2012 edition:**

- Improve exposition in the proof that $\sqrt{2}$ exists.
- Minor improvements in style and exposition in numerous places.
- Put "derivative" into the index.
- Use only "well ordering property" to make logicians happy.
- Remove definition of "size" of a partition as it was never actually used.
- Several minor typos and grammar errors fixed.

You can download the LaTeX source for this old version if you want (but I recommend just using the current version).

**April 8th 2012 edition:**

- Make Proposition 7.3.6 (ii) clearer and make it just a one way implication. The other direction is contained in (i) and just made the statement somewhat ambiguous.
- Add figure 3.2 for continuity, which renumbers the popcorn function figure.
- Add a sentence about defining arithmetic on extended real numbers.
- Some small improvements to readability in places.
- A few style, grammar, and spelling fixes.
- Some small changes in spacing (lists) to make them slightly more compact, also fiddle with pagination manually on a few pages to get nicer page breaks, so pagination changed slightly.

You can download the LaTeX source for this old version if you want (but I recommend just using the current version).

**December 25th 2011 edition:**

- Fix the errata from the last revision.
- Add Proposition 7.3.6 for limits of sequences and an exercise to prove it. This caused a
**slight renumbering**in 7.3. - Add reference to uniform norm in definition of C([a,b])
- Several very minor grammar/style fixes.

You can download the LaTeX source for this old version if you want (but I recommend just using the current version).

**December 15th 2011 edition:**

- Fix the errata from the last revision.
- Add examples 7.2.22, and 7.2.23, which caused a
**slight renumbering in 7.2**: proposition 7.2.22 became 7.2.24 and 7.2.23 became 7.2.25, and corollary 7.2.24 became 7.2.26. - Add a proposition that every convergent sequence is Cauchy (7.4.2)
This caused a
**renumbering in 7.4**. Furthermore, I realized that if we just prove the Lebesgue covering lemma, the proof of "sequentially compact implies compact" is about the same length but split up and hopefully easier to read. And we prove another useful fact as a bonus for no extra effort. This caused some more renumbering in in 7.4. - Add examples 7.5.3 and 7.5.8, which caused a
**slight renumbering in 7.5**and it move section 7.6 up a page. - Make Proposition 7.2.11 into an if and only if and change the assumption on the intersection to $U_1 \cap U_2 \cap S \not= \emptyset$.
- Improve exposition in a number of places in chapter 7.
- Use $\lambda$ instead of $\iota$ for an index for the arbitrary unions and intersections.
- Fix and improve the proof of proposition 7.2.13.
- Make the statement of 7.2.10 more precise.
- Proof of proposition 7.2.21 is much simpler, no need for unions.
- Add note about open sets being unions of balls, now that we don't use it in any proof. It is a useful thing to mention.
- Use proposition 7.3.7 in the proof of 7.3.8 to improve exposition.
- Slightly improve definition of compactness.
- Add exercises 7.2.12-14, 7.3.9-11, 7.4.11-14, 7.5.10, 3.1.11, 3.2.13.
- 7.6 probably only takes 1 lecture, unless one also does the examples from 6.3, so revise the number of lectures estimate.
- Reformulate exercise 7.6.2 to explicitly show that F(s,f(s)) is continuous as we use that later.
- Use x and y instead of t and x in section 7.6 to make it consistent with 6.3.
- Use $C(I,\mathbb{R})$ for real valued continuous functions on I rather than just C(I).

- In subsection 1.2.3, add the elementary proposition 1.2.8 (proof left to
reader) about suprema and existence of x arbitrarily close. This led to
**renumbering**as definition 1.2.8 became definition 1.2.9. - In exercise 1.2.2: change $t > 0$ to $t \geq 0$ which makes it more natural.
- Mark exercise 2.5.7 as challenging.
- In corollary 3.1.12 (iv) no need to require that $g(c) \neq 0$.
- In exercise 4.2.4, require that the sequence converges to c.
- Provide alternative reverse directions of Lemma 3.1.7 and Proposition 3.2.2 (ii) as exercises 3.1.11 and 3.2.13
- Several improvements in grammar and style, and several minor typos.

**November 18th 2011 edition:**

**Add Chapter 7 on Metric spaces**. The pagination of Chapters 1-6 did not change. There are 192 pages to the book now, with 286 exercises.- Fix the errata from the last revision.
- Add exercises 1.1.9, 2.1.18, 3.1.10, 3.3.8, 4.1.11, 4.2.9, 4.2.10, 5.1.8-11, 5.2.14, 5.2.15, 6.1.11 (and of course all the new metric space exercises)
- Add example 6.3.6
- In exercise 5.3.6, require that it be done by FTC, not by mean value theorem as a previous similar exercise.
- In Theorem 6.2.2, change the domain to be arbitrary, there is no need to only consider intervals.
- Many small improvements and fixes in both exposition and style, mainly in chapters 3 through 6.
- Improved definition of uniform convergence
- In section 6.2, add in some minor omitted details
- Revised some of the "number of lectures estimates" to match somewhat better what what I actually do. It is of course still very approximate and depends on the level of the students, etc...

**October 16th 2011 edition:**

- Fix the errata from the last revision.
- Reorder subsection 1.2.3. The "extended reals" are moved to end of the
subsection. It is really mostly an optional thing, we almost
never use it (I mention
this fact as well). This resulted in a
**slight renumbering**: Definition 1.2.6 became Definition 1.2.8, and Propositions 1.2.7 and 1.2.8 are Propositions 1.2.6 and 1.2.7 respectively. - Add exercises 1.1.8, 1.3.6, 2.1.17, 2.3.11, 2.3.12, 2.5.8, 2.5.9, 2.5.10, 3.1.9
- Exercises are now set in slightly smaller font.
- Add footnote to Proposition 1.3.7 for the interested reader that bounded is not necessary if one uses the extended reals (and add as exercise).
- Explicitly define in 1.1 the word "bounded" for ordered sets to mean bounded from above and below.
- In exercise 1.1.7: add "nonempty" hypothesis to avoid an easy way out via a technicality.
- Add note about 0x=0 when introducing fields.
- Improve Figure 2.1.
- Rework and improve the proof that the reals are uncountable.
- Rewrite Exercise 2.4.3 (and fix the erratum that density, or some other form of archimedean property, is needed; density is natural to use here). Also marked as challenging.
- Make the distinction of "claim" and "proof" more explicit in all examples by adding "Proof:" if needed. Some examples seemed to be causing trouble for some students not accustomed to reading mathematics.
- Improve wording slightly in a number of places, for example in the definition of continuity.
- Lots of minor improvements in style and many minor typo fixes, especially chapters 1, 2, and beginning of 3.

**April 26th 2011 edition:**

- Reword remark 2.4.6 so that it is not interpreted wrongly (thanks to Frank Beatrous)
- Add exercises 2.5.6, 2.5.7, 4.2.8, 6.2.12
- Many minor typo fixes and clarifications.
- Fix the errata from the last revision.

**February 28th 2011 edition:**

- The proof of Prop 1.2.2 actually proved a stronger result, that is we only need to assume that $0 \leq x \leq \epsilon$ instead of $0 \leq x < \epsilon$. So state it as such (the "weaker" statement has a simpler proof by taking $x=\epsilon$).
- Add exercise 6.2.11.
- Some minor grammar and cosmetic fixes.
- Fix the errata from the last revision.

**December 26th 2010 edition:**

- Use $\{ x \in A : P(x) \}$ for set building notation instead of $\{ x \in A \mid P(x) \}$ to avoid visual conflict with absolute values.
- On page 11, fix "So let us assume that $x \in A \cap (B \cup C)$" which should be "So let us assume that $x \in A \setminus (B \cup C)$". Thanks to Dan Stoneham.
- On page 80, first line of subsection 3.1.2 c is of course a cluster point of S, not A.
- Improve Figure 3.1 near the origin, it looks a lot cleaner now.
- Some grammatical and cosmetic fixes.

**November 1st 2010 edition:**

- In exercise 1.3.4, the functions f and g are of course bounded as in proposition 1.3.7. (Glen Pugh)
- Fix proof of proposition 2.2.5 part iii was off if $y=0$. (Glen Pugh)
- Fix the definition of the function defined in the beginning of Example 6.2.3, to match the graph on Figure 6.3.
- In example 4.2.10, finish the argument showing that f' is not continuous at zero (and leave the actual computation to the student).
- Some grammatical and cosmetic fixes.

**October 3rd 2010 edition:**

- Slightly modify Example 0.3.14 to fix a typo (Thanks to Glen Pugh).
- Fix typo in the statement of Proposition 0.3.16 (Thanks to Glen Pugh).
- Fix the typo in the explanatory sentence in Exercise 0.3.4 part b) (Thanks to Glen Pugh).
- Add note that bounded does not imply convergent for sequences.
- Explicitly mention order when introducing subsequences.
- Some clarifications.
- Some grammar and spelling fixes.

**September 6th 2010 edition:**

The numbering of theorems, propositions, exercises, etc.. has not changed at all. The pagination has essentially not changed either.

- On bottom of page 11, the counter-example for swapping intersection and union didn't work due to a typo. The set is defined by $\{ k \in \mathbb{N} \mid mk < n \}$. Thanks to Glen Pugh.
- Few extra internal hyperlinks.
- Some minor clarifications and grammar fixes.

I forgot to make an archive of LaTeX sources for the September 6th edition.

**August 12th 2010 edition:**

The numbering of theorems, propositions, exercises, etc.. has not changed at all. The pagination has essentially not changed either.

- Exercise 5.2.11 mentioned as a side note that the Thomae function is "everywhere discontinuous," which is a typo. It is discontinuous on the rational numbers which is a dense set. (Thanks to an anonymous reader for noticing)
- More hyperlinks
- Few more indexed terms
- Add note about Leibniz for product rule
- Some clarifications and grammar/typo fixes

**July 15th 2010 edition:**

The numbering of theorems, propositions, exercises, etc.. has not changed at all. The pagination has essentially not changed either.

- Use microtype package with pdflatex for nicer looking output and better line breaks.
- Fix some hyper linking issues, and make the use the whole name of the object as the link target rather than just the number.
- Some minor grammar fixes.

**June 23rd 2010 edition:**

The numbering of theorems, propositions, exercises, etc.. has not changed at all. The pagination has not changed either.

All the errata fixed were found thanks to Jana Maříková.

- p.13, the denominator should be $1-c^{n+1} + (1-c)c^{n+1}$
- p.15, Def. 0.3.18: f and g were switched in the definition.
- p.21, Def. 1.1.2: the definition of lower bound was missing
- p.22, Def. 1.1.3: need to assume that E is nonempty
- p.61, $y=\frac{b_k - a_k}{2}$ should be $y=\frac{a_k + b_k}{2}$.
- p.66, "b:=liminf x
_{n}" should be "b:=limsup x_{n}" - p.87, "Fix $c \in (0,\infty)$." should be "Fix $c \in (-\infty , \infty)$."
- p.87, "x" was missing the lower index "n"
- p.102, Exercise 3.4.5: A and B are assumed to be intervals with nonempty intersection for the exercise to be possible. Add second part to find a counterexample where A and B are disjoint.
- p.105, numerator of second fraction should be "(f(x)+g(x))-(f(c)+g(c))"
- p.107: equality was missing between "...(g(x)-g(c))" and "u(g(x))..."
- Some other minor typo and cosmetic fixes.
- Using new texlive, which outputs PDF 1.5, which is significantly smaller.
- Underline links, the underlines should not appear in printed output.

**April 8th 2010 edition:**

The numbering of theorems, propositions, exercises, etc.. has not changed at all. The pagination has not changed either.

- Add exercises 0.3.20 and 1.1.7.
- Minor clarifications in places.
- Lots of minor typo and grammar fixes.

**December 23rd 2009 edition:**

- Fixed typo in proof of Proposition 5.2.4 (there were a bit too many alphas around)
- The proof of Picard's theorem does not require us to assume that an
interval of radius 2α around y
_{0}is in J. [y_{0}-α,y_{0}+α] suffices. - Add hint to Exercise 5.2.2.
- Note use of proposition 5.1.8 in example 5.1.12 and explicitly allow its use in the exercises (to avoid confusion).
- Fixed minor typos and grammar mistakes.

**December 11th 2009 edition:**

First version. You can download the LaTeX source for this old version if you want (but I recommend just using the current version).