# Basic Analysis: Introduction to Real Analysis: Changes

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.

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.

1. Add optional section 1.5 on decimal expansion. This makes the hard exercise 1.4.3 obsolete, so replace with another exercise.
2. Add optional section 2.6 on more topics on series
3. Add optional section 3.5 on limits at infinity
4. Add optional section 3.6 on monotone functions and continuity
5. Add optional section 4.4 on inverse function theorem
6. Add optional section 5.4 on the logarithm and the exponential
7. Add optional section 5.5 on improper integrals
8. Add Proposition 5.1.13 and another example 5.1.14 to show integrability directly.
9. 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.
10. 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.
11. 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
12. In exercise 7.3.5, onto is not needed, and might confuse the student, so remove the hypothesis.
13. Mark the dependence on x0 on the Taylor polynomial and remainder.
14. 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.
15. In references to famous mathematicians link the Wikipedia article. Also we completely lacked such a footnote for Cantor.
16. 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.
17. Fix the errata from the last revision (In particular exercise 7.3.10 was replaced).
18. Minor English and style fixes.

December 16th 2012 edition:

1. Fix the errata from the last revision.
2. Footnote counter is reset per page as is usual, so that we don't go into weird footnote symbols.
3. In Taylor's theorem proof make it explicit that M and hence c depends on x and x0. (Thanks to Sonmez Sahutoglu for the suggestion)
4. Add exercises 3.3.9, 4.3.4, 5.3.10, 5.3.11

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

October 1st 2012 edition:

1. Improve exposition in the proof that $\sqrt{2}$ exists.
2. Minor improvements in style and exposition in numerous places.
3. Put "derivative" into the index.
4. Use only "well ordering property" to make logicians happy.
5. Remove definition of "size" of a partition as it was never actually used.
6. 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:

1. 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.
2. Add figure 3.2 for continuity, which renumbers the popcorn function figure.
4. Some small improvements to readability in places.
5. A few style, grammar, and spelling fixes.
6. 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:

1. Fix the errata from the last revision.
2. Add Proposition 7.3.6 for limits of sequences and an exercise to prove it. This caused a slight renumbering in 7.3.
3. Add reference to uniform norm in definition of C([a,b])
4. 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:

1. Fix the errata from the last revision.
2. 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.
3. 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.
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.
5. Make Proposition 7.2.11 into an if and only if and change the assumption on the intersection to $U_1 \cap U_2 \cap S \not= \emptyset$.
6. Improve exposition in a number of places in chapter 7.
7. Use $\lambda$ instead of $\iota$ for an index for the arbitrary unions and intersections.
8. Fix and improve the proof of proposition 7.2.13.
9. Make the statement of 7.2.10 more precise.
10. Proof of proposition 7.2.21 is much simpler, no need for unions.
11. 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.
12. Use proposition 7.3.7 in the proof of 7.3.8 to improve exposition.
13. Slightly improve definition of compactness.
14. Add exercises 7.2.12-14, 7.3.9-11, 7.4.11-14, 7.5.10, 3.1.11, 3.2.13.
15. 7.6 probably only takes 1 lecture, unless one also does the examples from 6.3, so revise the number of lectures estimate.
16. Reformulate exercise 7.6.2 to explicitly show that F(s,f(s)) is continuous as we use that later.
17. Use x and y instead of t and x in section 7.6 to make it consistent with 6.3.
18. Use $C(I,\mathbb{R})$ for real valued continuous functions on I rather than just C(I).
The following changes are suggestions from Paul Vojta:
1. 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.
2. In exercise 1.2.2: change $t > 0$ to $t \geq 0$ which makes it more natural.
3. Mark exercise 2.5.7 as challenging.
4. In corollary 3.1.12 (iv) no need to require that $g(c) \neq 0$.
5. In exercise 4.2.4, require that the sequence converges to c.
6. Provide alternative reverse directions of Lemma 3.1.7 and Proposition 3.2.2 (ii) as exercises 3.1.11 and 3.2.13
7. Several improvements in grammar and style, and several minor typos.

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

November 18th 2011 edition:

1. 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.
2. Fix the errata from the last revision.
3. 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)
5. In exercise 5.3.6, require that it be done by FTC, not by mean value theorem as a previous similar exercise.
6. In Theorem 6.2.2, change the domain to be arbitrary, there is no need to only consider intervals.
7. Many small improvements and fixes in both exposition and style, mainly in chapters 3 through 6.
8. Improved definition of uniform convergence
9. In section 6.2, add in some minor omitted details
10. 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...

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

October 16th 2011 edition:

1. Fix the errata from the last revision.
2. 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.
3. 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
4. Exercises are now set in slightly smaller font.
5. 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).
6. Explicitly define in 1.1 the word "bounded" for ordered sets to mean bounded from above and below.
7. In exercise 1.1.7: add "nonempty" hypothesis to avoid an easy way out via a technicality.
9. Improve Figure 2.1.
10. Rework and improve the proof that the reals are uncountable.
11. 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.
12. 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.
13. Improve wording slightly in a number of places, for example in the definition of continuity.
14. Lots of minor improvements in style and many minor typo fixes, especially chapters 1, 2, and beginning of 3.

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

April 26th 2011 edition:

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

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

February 28th 2011 edition:

1. 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 $x=\epsilon$).
3. Some minor grammar and cosmetic fixes.
4. Fix the errata from the last revision.

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

December 26th 2010 edition:

1. 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.
2. 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.
3. On page 80, first line of subsection 3.1.2 c is of course a cluster point of S, not A.
4. Improve Figure 3.1 near the origin, it looks a lot cleaner now.
5. Some grammatical and cosmetic fixes.

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

November 1st 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. Fix proof of proposition 2.2.5 part iii was off if $y=0$. (Glen Pugh)
3. Fix the definition of the function defined in the beginning of Example 6.2.3, to match the graph on Figure 6.3.
4. In example 4.2.10, finish the argument showing that f' is not continuous at zero (and leave the actual computation to the student).
5. Some grammatical and cosmetic fixes.

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

October 3rd 2010 edition:

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

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

September 6th 2010 edition:

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

1. 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.
3. 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.

1. 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)
3. Few more indexed terms
5. Some clarifications and grammar/typo fixes

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

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.

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

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 xn" should be "b:=limsup xn"
• 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.

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

April 8th 2010 edition:

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

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

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

December 23rd 2009 edition:

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

1. Fixed typo in proof of Proposition 5.2.4 (there were a bit too many alphas around)
2. The proof of Picard's theorem does not require us to assume that an interval of radius 2α around y0 is in J. [y0-α,y0+α] suffices.
3. Add hint to Exercise 5.2.2
4. Note use of proposition 5.1.8 in example 5.1.12 and explicitly allow its use in the exercises (to avoid confusion).
5. Fixed minor typos and grammar mistakes

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

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).