[Go to the Basic Analysis home page]

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.

**June 10th 2020 edition, Version 5.3 (edition 5, 3rd update):**

- Remove the note before exercise 7.2.12. It is obsolete and could be confusing as we have already defined "subspace topology" above.
- Note after Definition 0.3.13 that $R(f)=f(A)$.
- Add very small example to composition of functions after Definition 0.3.18.
- Add a couple of sentences about why existence of a bijection is an equivalence relation.
- Note why the only set with cardinality of 0 is the empty set.
**Exercise 5.2.13 part d) replaced, part e) adds a hypothesis (differentiable at 0)**(to fix errata)- Many minor language and style improvements as well as some minor clarifications.
- Fix the errata from the last revision.

**May 15th 2019 edition, Version 5.2 (edition 5, 2nd update):**

- Several minor grammar and style fixes.
- Fix the errata from the last revision.
**Exercise 3.6.10 changed slightly due to an erratum**: a hypothesis that the function is bounded is necessary.

**Oct 11th 2018 edition, Version 5.1 (edition 5, 1st update):**

- When introducing power series, explicitly mention $0^0=1$.
- Links are now https.
- Fix the errata from the last revision.

**May 7th 2018 edition, Version 5.0 (edition 5, 0th update):**

The motivation for this revision is to improve readability of existing material
rather than adding much new material.
To this end,
**39 new figures** were added (so 65 total)
there are several new examples,
as well as reorganizing and expanding explanations throughout.
Furthermore,
**99 new exercises** were added bringing the total to 528
(plus two had to be replaced).

A **List of Notations** is added at the back, giving a description and
a page number for the most relevant definition or use for the notations used
in the book.

There are the following more major additions:
**A short new subsection** in 0.3 on relations.
**Two new subsections** in 6.2 on limits of derivatives, and on power series.
I always felt like chapter 6 ends too abruptly if 6.3 is not covered. This
adds a nice simpler application of swapping of limits with much easier proof
than Picard.
**A short new subsection** on limits of functions in 7.5, as this is
really used in chapter 8 of volume II.
**Section 4.3 was expanded** with a discussion on Taylor series, as
well as the second derivative test.
Throughout the book, some material that was in remarks, examples, and exercises
but was used often, was formalized into several new propositions.

Some exercises, examples, propositions were added, some theorems became
propositions, a few definitions, theorems, propositions, lemmas, corollaries,
and examples in 0.3, 1.4, 2.5, 3.4, 4.1, 4.2, 5.2, 7.1, 7.2, 7.3, 7.4 were
renumbered. Figure numbers have changed.
**Existing exercise numbers are the same, except:
Exercises 2.5.1, 7.2.5 were replaced.
7.2.12 asks for the reverse implication (that was an erratum, as this was
already proved in the text).
In exercise 3.3.11 we require that the example is continuous.
Exercise 4.4.6 was simplified very slightly (the original was a typo).
Due to new material, exercise 3.6.2 now asks for more, and exercise
6.2.7 is easier. Exercise 7.5.9 is easier with a new proposition.
A couple of other exercises had errata fixed (0.3.5, 1.4.3,
5.4.2, 7.1.5).**
Other than this, the new edition is essentially backward compatible as usual.

A detailed list of changes:

- Identify the book as Volume I on the title page, and refer to Volume II in the introduction.
- In the PDF the pages have been made slightly longer so that we can lower the page count to save some paper.
- To be more consistent with what is a Theorem and what is a Proposition, demote Theorems 2.1.10, 2.3.5, 2.3.7, 3.4.6, 5.2.2 to Propositions. Also change Theorem 4.2.2 to a Lemma as that's more appropriate. Numbering didn't change.
- Change the look of the figures to match the Volume II and to better visually distinguish them from the surrounding text.
- Change the "basic analysis result" to $x \leq \epsilon$ for all $\epsilon > 0$ implies $x \leq 0$. This better fits the mantra that in analysis we prove inequalities, and separates out the idea that to show $x=0$ one proves $x \leq 0$ and $x \geq 0$.
- Add a short paragraph about naming of Theorem vs Proposition vs Lemma vs Corollary, to answer a common question.
- Add a subsection on relations, equivalence relations, and equivalence classes. This renumbers the following propositions, definitions, etc.
- Add figure for the sets $S$ and $T$ in 0.3
- Add figure for direct/inverse images in section 0.3.
- Add figure for showing ${\mathbb{N}}^2$ is countable.
- Add exercises 0.3.21, 0.3.22, 0.3.23, 0.3.24, 0.3.25.
- Add figure for least upper bound definition.
- Add note about uniqueness of sups and infs.
**In proposition 1.1.8, add the two very commonly used properties as parts (vi) and (vii).**- Add explicitly proposition 1.1.11 about an ordered field with LUB property also having GLB property.
- Add link to Dedekind's Wikipedia page.
- In exercise 1.1.6, removed the "In particular, $A$ is infinite". There is no point in going into the distinction and it just confuses students.
- Add exercises 1.1.11, 1.1.12, 1.1.13, 1.1.14.
- Add footnote on impossibility of tuned pianos and rational roots
**In proposition 1.2.2 simplify matters by changing the statement to not assume that $x \geq 0$.**The original statement is given in the paragraph below as a remark.- Add figure to proof of the density of $\mathbb Q$ in section 1.2.
- Add exercise 1.2.14, 1.2.15, 1.2.16, 1.2.17.
- Change title of 1.3 to include "bounded functions".
- Add figure for a bounded function, its supremum and its infimum in section 1.3.
- Add exercises 1.3.8, 1.3.9.
- Add Proposition 1.4.1 (which moves Theorem 1.4.1 to 1.4.2), which is the characterization of intervals that we often use later, so better to formalize it. Proof is still an exercise.
- We never defined/open closed for unbounded intervals, although later on we make a big deal about a closed and bounded interval. To be more in line with general usage, define what "unbounded closed" and "unbounded open" intervals.
- In exercise 1.4.6, be more explicit about what the intersection of closed intervals is, and explicitly mention boundedness. That is, say the intersection is $\cap_{\lambda \in I} [a_\lambda,b_\lambda]$.
- Add exercise 1.4.10.
- In Proposition 1.5.1, add the inequalities for all representations as well, since we use these facts later. Also add the detail of the proof as it is perhaps not as obvious to every reader.
- Mark exercise 1.5.6 as challenging and add a longer hint. The real tricky part is to get a bijection rather than two injections which is easier.
- Add exercise 1.5.8, which is really required in the proof, so that we do not require things from chapter 2. Be more explicit about its use in the proof.
- Add figure on cantor diagonalization in section 1.5.
- Add more detail in proof of Proposition 1.5.3, to see how we use the unique representation.
- Add exercises 1.5.7, 1.5.9
- Add a very short example of a tail of a sequence in 2.1.
- Add a diagram to proof of Proposition 2.1.15.
- Simplify the proof of squeeze lemma as suggested by Atilla YÄ±llmaz.
- Add example of showing $n^{1/n}$ going to 1 as a more subtle example of the use of the ratio test.
- Simplify/symmetrize the proof of product of limits is the limit of the product. (Thanks to Harold Boas)
- Show the convergence/unboundedness of $\{ c^n \}$ in a somewhat a more elementary way without Bernoulli's inequality. (Thanks to Harold Boas)
- Add exercises 2.2.13, 2.2.14, 2.2.15, 2.2.16.
- Add two figures in 2.3 for liminf and limsups, one for a random example, and one for the given example.
- Expand the discussion of infinite limits and liminf/limsup for unbounded sequences. Add a proposition about unbounded monotone sequences, and a proposition connecting the definition of liminf/limsup to the previous definition for bounded sequences.
- Add exercises 2.3.15, 2.3.16, 2.3.17, 2.3.18, 2.3.19.
- Add figure to the example of geometric series with 1/2.
- Make the geometric series into a Proposition as we use it quite a bit. Also use geometric series as an example for the divergence if terms do not go to 0, that is when $r \notin (-1,1)$.
- Mention the ``infinite triangle inequality'' in text in 2.5, I always do in class. These two things renumber the subsequent examples, propositions, etc. in 2.5
**Replace exercise 2.5.1**. The exercise was proved in example 0.3.8 and already used previously.- Add exercises 2.5.14, 2.5.15, 2.5.16, 2.5.17.
- Add a sentence and notation to the figure about possible non-convergence at the endpoints of the radius of convergence.
- Add exercises 2.6.13, 2.6.14.
- Add a note and a footnote on the other common notations for the various limits of restrictions.
- Add Corollary after 3.1.12 for the absolute value, which shifts the numbering of propositions and examples by one in 3.1.
- Add exercises 3.1.15, 3.1.16.
- Expand example 3.2.10 a little bit, and add a figure for the example.
- Add exercises 3.2.17, 3.2.18, 3.2.19.
- Add figure for definition of absolute minimum and maximum.
- Add corollary 3.3.12 whose proof is the existing exercise 3.3.7.
**In exercise 3.3.11 add the missing continuity hypothesis**. Otherwise the exercise is too easy (it is already easy).- Add exercise 3.3.14, 3.3.15, 3.3.16, 3.3.17.
- Swap examples 3.4.2 and 3.4.3, they make a lot more sense in that order.
- In Section 3.4 add a very short application of the continuous extension.
- Add figure for the idea of the proof of the product rule, that is, a picture of the identity given as hint.
- Add exercises 3.4.15, 3.4.16, 3.4.17.
- Add exercise 3.5.9.
- Strengthen Proposition 3.6.2 to include limits at infinity,
which means that
**Exercise 3.6.2 asks for a bit more**since two new statements must be proved. - Add exercises 3.6.12, 3.6.13, 3.6.14, 3.6.15.
- Actually prove the use of intermediate value theorem in proof of corollary 3.6.3.
- Add figure to Example 3.6.5.
- Add figure to Example 3.6.7.
- Add examples 4.1.3, 4.1.4 which moves everything down a number in 4.1.
- Add link to Schwarz and Bunyakovsky and give a short note on the name in a footnote.
- Add exercises 4.1.13, 4.1.14, 4.1.15.
- Reorganize the proof of Mean value theorem a little bit, add some motivation for the proof, and move the figure up earlier as it gives an idea for the proof.
- Make Example 4.2.8 into a Proposition since that's what it really is. Then we can refer to it rather than the exercise that proves it later.
- The proof of exercise 4.2.9 was a little too challenging. In essence one reproves Cauchy's mean value theorem anyway, so add that as a theorem, and add an exercise to prove it. This causes some renumbering in 4.2.
- Add a proposition about extension of derivatives to the boundary as that is a in fact quite useful and has a very quick and straightforward proof which is left as exercise.
- Add small note about measuring speed with aircraft and mean value theorem.
- Add some motivation to the proof of Darboux's theorem, and add a figure.
- Add exercises 4.2.13, 4.2.14, 4.2.15.
- Add two figures for Taylor's theorem section (4.3).
- Mention Taylor series and connection to power series in 4.3.
- Add quick application of Taylor's theorem to prove second derivative test. Proposition 4.3.3.
- Add exercises 4.3.9, 4.3.10, 4.3.11.
- Rewrite proof of Lemma 4.4.1, and use clearer variable names.
- Add figure to Example 4.4.5.
**Modify Exercise 4.4.6 very slightly**, replace ``interval'' with ``open interval''. The distinction is irrelevant for how one proves it and considering other types of intervals makes the proof longer.- Add figure to the proof of Proposition 5.1.7.
- Add figure to Proposition 5.1.10.
- Add figure to Example 5.1.12.
- Add proposition on the sub/super additivity as proposition 5.2.5, so all other propositions, theorems, and lemmas shift by one in 5.2.
- In the monotonicity proposition, state it for upper and lower integrals as well, we prove that anyway, it fits better with the style of exposition in this book, and it can be useful in proofs.
- Add proposition for the integrability of monotone functions. We use this later, it is better to just refer to a proposition than an exercise, and it is also genuinely useful.
- Add exercise 5.2.17.
- Add figure to proof of the fundamental theorem of calculus in 5.3.
- Add remarks about other definitions of logarithm and the exponential, and about the uniqueness and existence following from a subset of the given conditions.
- Add exercise 5.4.11.
- Improve the exposition of the summability of the sinc function in example 5.5.12 and add another figure to the example to show the bound.
- Add figure for integral test for series in 5.5.
- Add figure to example 6.1.4.
- Add figure to definition uniform convergence in 6.1.
- Add exercise 6.1.12, 6.1.13, 6.1.14.
- Add subsection to 6.2 on swapping of limit of functions and derivatives
for continuously differentiable functions.
**This makes exercise 6.2.7 much easier**as we essentially do the main bit as a theorem. There is a new figure in this subsection. - Add subsection to 6.2 on convergence, differentiation, and integration of power series.
- Change hint in 6.2.1 to be simpler, $|x|^{1+1/n}$ works but it is a bit messy to prove all the details.
- Add exercises 6.2.15, 6.2.16, 6.2.17, 6.2.18, 6.2.19, 6.2.20, 6.2.21.
- Add remark about weaker solutions to ODEs using the integral equation.
- Use the more common interior notation in 6.3, and in 7.6.
- Add exercises 6.3.7, 6.3.8, 6.3.9.
- Improve triangle inequality figure in 7.1.
- Add example of complex numbers to 7.1, and an example of a sphere, that renumbers the rest of the examples and propositions in 7.1.
- Add exercises 7.1.9, 7.1.10, 7.1.11, 7.1.12, 7.1.13.
- Improve the open set figure in 7.2.
- Add Propositions 7.2.11 and 7.2.12 that codify some of the subspace topology things we keep using. This renumbers the rest of the definitions, examples, and propositions in 7.2.
- Simplify proof of Proposition 7.2.15, as the conclusion was already proved in exercise in 1.4, and is formalized in Proposition 1.4.1.
**Replace exercise 7.2.5**, the conclusion was already proved in exercise 1.4.3 (in more generality, in fact).**In Exercise 7.2.12**the implication goes the other way (erratum in earlier versions), as is needed in the text.- Add figures to Propositions 7.2.9, 7.2.13, 7.2.15, and 7.2.26.
- Add $(0,\infty)$ and $[0,\infty)$ as an examples of an open and closed sets in ${\mathbb{R}}$ to Example 7.2.5.
- Add footnote about empty sets and connectedness.
- Add exercises 7.2.15, 7.2.16, 7.2.17, 7.2.18.
- Add figure to definition of convergence in 7.3.
- Add example to 7.3 of $C([0,1],{\mathbb{R}})$ where convergence is the uniform convergence. This renumbers the following examples, propositions, etc.
- Add remark that pointwise convergence does not come from a metric.
- Add example for convergence in the complex numbers.
- Add exercises 7.3.13, 7.3.14.
- Add an example (in fact a set of 4 examples) of compact and noncompact sets on the real numbers in 7.4. This again renumbers the remaining propositions, etc.
- Add proposition that $C([a,b],{\mathbb R})$ is a complete metric space.
- Add proposition that a closed subset of a complete metric space is complete, that is used later.
- Add remark at the end of 7.4 about Cauchy depending on the actual metric and not just on the topology, along with an exercise working through the counterexample.
- Add an example for the Lebesgue covering lemma, finding a $\delta$ for a cover.
- Add figures to proof of Proposition 7.4.9, Lebesgue covering lemma, and Theorem 7.4.11.
- Add exercises 7.4.17, 7.4.18, 7.4.19, 7.4.20.
- Add figure for Lemma 7.5.7.
- Add a proposition 7.5.12 on continuity of functions defined by integration. Makes exercise 7.5.9 simpler, but it seemed to that most students missed the subtlety, and we use this result later a few times.
- Add exercises 7.5.11, 7.5.12, 7.5.13, 7.5.14, 7.5.15, 7.5.16, 7.5.17, 7.5.18.
- Make notation more in line with the rest of the chapter in 7.6.
- Move all exercises to the Exercises subsection 7.6 to be consistent with the rest of the book.
- Add exercise 7.6.11.

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

**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 examples 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 dictionary.
- 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 inequality 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 sequences. 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 explicit 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 Proposition 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 Picard's 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 or PDF for this old version if you want (but I recommend just using the current version).

**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.

**October 7th 2013 edition:**

- Fix the errata from the last revision.
- Fix some grammar/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 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 $2\alpha$ around $y_0$ is in $J$. $[y_0-\alpha,y_0+\alpha]$ 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).