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 This file is a draft of the new changes for http://www.jirka.org/ra/changes.html
 
 <p>
-<b>??? ??th 2018 edition, Version 5.0 (edition 5, 0th update):</b>
-<p>
-The motivation for this revision is to improve readability of existing material
-rather than adding much new material.
-To this end,
-<b>39 new figures</b> were added (so 65 total) <!-- 26 in last edition -->
-there are several new examples,
-as well as reorganizing and expanding explanations throughout.
-Furthermore,
-<b>99 new exercises</b> were added bringing the total to 528 total
-(plus two had to be replaced). <!-- 429 in last edition -->
-<p>
-A <b>List of Notations</b> 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.
-<p>
-There are the following more major additions:
-<b>A short new subsection</b> in 0.3 on relations.
-<b>Two new subsections</b> 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.
-<b>A short new subsection</b> on limits of functions in 7.5, as this is
-really used in chapter 8 of volume II.
-<b>Section 4.3 was expanded</b> 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.
-<p>
-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.
-<b>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),
-and 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).</b>
-Other than this, the new edition is essentially backward compatible as usual.
-
-<p>
-A detailed list of changes:
-<ol>
-<li>Identify book as Volume I on the title page, and refer to Volume II
-in the introduction.
-<li>In the PDF the pages have been made slightly longer so that we can lower
-the page count to save some paper.
-<li>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.
-<li>Change the look of the figures to match the Volume II and
-to better visually distinguish them from the surrounding text.
-<li>Change the "basic analysis result" to $x \leq \epsilon$
-for all $\epsilon &gt; 0$ 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 $x \geq 0$.
-<li>Add a short paragraph about naming of Theorem vs Proposition vs Lemma vs
-Corollary, to answer a common question.
-<li>Add a subsection on relations, equivalence relations,
-and equivalence classes.  This renumbers the following
-propositions, definitions, etc...
-<li>Add figure for the sets $S$ and $T$ in 0.3
-<li>Add figure for direct/inverse images in section 0.3.
-<li>Add figure for showing ${\mathbb{N}}^2$ is countable.
-<li>Add exercises 0.3.21, 0.3.22, 0.3.23, 0.3.24, 0.3.25.
-<li>Add figure for least upper bound definition.
-<li>Add note about uniqueness of sups and infs.
-<li><b>In proposition 1.1.8, add the two very commonly used properties
-as parts (vi) and (vii).</b>
-<li>Add explicitly proposition 1.1.11 about an ordered field with
-LUB property also having GLB property.
-<li>Add link to Dedekind's Wikipedia page.
-<li>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.
-<li>Add exercises 1.1.11, 1.1.12, 1.1.13, 1.1.14.
-<li>Add footnote on impossibility of tuned pianos and rational roots
-<li><b>In proposition 1.2.2 simplify matters by changing the statement to not
-assume that $x \geq 0$.</b>  The original statement is given in the paragraph
-below as a remark.
-<li>Add figure to proof of the density of $\mathbb Q$ in section 1.2.
-<li>Add exercise 1.2.14, 1.2.15, 1.2.16, 1.2.17.
-<li>Change title of 1.3 to include "bounded functions".
-<li>Add figure for a bounded function, its supremum and its infimum
-in section 1.3.
-<li>Add exercises 1.3.8, 1.3.9.
-<li>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.
-<li>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.
-<li>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 $\cap_{\lambda I} [a_\lambda,b_\lambda]$.
-<li>Add exercise 1.4.10.
-<li>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.
-<li>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.
-<li>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.
-<li>Add figure on cantor diagonalization in section 1.5.
-<li>Add more detail in proof of Proposition 1.5.3, to see how we use the
-unique representation.
-<li>Add exercises 1.5.7, 1.5.9
-<li>Add a very short example of a tail of a sequence in 2.1.
-<li>Add a diagram to proof of Proposition 2.1.15.
-<li>Simplify the proof of squeeze lemma as suggested by Atilla Yıllmaz.
-<li>Add example of showing $n^{1/n}$ going to 1 as a more subtle example of
-the use of the ratio test.
-<li>Simplify/symmetrize the proof of product of limits is the limit of the
-product.  (Thanks to Harold Boas)
-<li>Show the convergence/unboundedness of $\{ c^n \}$ in a somewhat 
-a more elementary way without Bernoulli's inequality.
-(Thanks to Harold Boas)
-<li>Add exercises 2.2.13, 2.2.14, 2.2.15, 2.2.16.
-<li>Add two figures in 2.3 for liminf and limsups, one for a random example,
-and one for the given example.
-<li>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.
-<li>Add exercises 2.3.15, 2.3.16, 2.3.17, 2.3.18, 2.3.19.
-<li>Add figure to the example of geometric series with 1/2.
-<li>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 $r \notin (-1,1)$.
-<li>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
-<li><b>Replace exercise 2.5.1</b>.  The exercise was proved in example 0.3.8
-and already used previously.
-<li>Add exercises 2.5.14, 2.5.15, 2.5.16, 2.5.17.
-<li>Add a sentence and notation to the figure about possible
-non-convergence at the endpoints of the radius of convergence.
-<li>Add exercises 2.6.13, 2.6.14.
-<li>Add a note and a footnote on the other common notations for the various
-limits of restrictions.
-<li>Add Corollary after 3.1.12 for the absolute value, which shifts the
-numbering of propositions and examples by one in 3.1.
-<li>Add exercises 3.1.15, 3.1.16.
-<li>Expand example 3.2.10 a little bit, and add a figure for the example.
-<li>Add exercises 3.2.17, 3.2.18, 3.2.19.
-<li>Add figure for definition of absolute minimum and maximum.
-<li>Add corollary 3.3.12 whose proof is the existing exercise 3.3.7.
-<li><b>In exercise 3.3.11 add the missing continuity hypothesis</b>.
-Otherwise the exercise is too easy (it is already easy).
-<li>Add exercise 3.3.14, 3.3.15, 3.3.16, 3.3.17.
-<li>Swap examples 3.4.2 and 3.4.3, they make a lot more sense in that order.
-<li>In Section 3.4 add a very short application of the continuous extension.
-<li>Add figure for the idea of the proof of the product rule, that is,
-a picture of the identity given as hint.
-<li>Add exercises 3.4.15, 3.4.16, 3.4.17.
-<li>Add exercise 3.5.9.
-<li>Strengthen Proposition 3.6.2 to include limits at infinity,
-which means that <b>Exercise 3.6.2 asks for a bit more</b>
-since two new statements must be proved.
-<li>Add exercises 3.6.12, 3.6.13, 3.6.14, 3.6.15.
-<li>Actually prove the use of intermediate value theorem in proof of
-corollary 3.6.3.
-<li>Add figure to Example 3.6.5.
-<li>Add figure to Example 3.6.7.
-<li>Add examples 4.1.3, 4.1.4 which moves everything down a number in 4.1.
-<li>Add link to Schwarz and Bunyakovsky and give a short note on the name
-in a footnote.
-<li>Add exercises 4.1.13, 4.1.14, 4.1.15.
-<li>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.
-<li>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.
-<li>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.
-<li>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.
-<li>Add small note about measuring speed with aircraft and mean value theorem.
-<li>Add some motivation to the proof of Darboux's theorem, and
-add a figure.
-<li>Add exercises 4.2.13, 4.2.14, 4.2.15.
-<li>Add two figures for Taylor's theorem section (4.3).
-<li>Mention Taylor series and connection to power series in 4.3.
-<li>Add quick application of Taylor's theorem to prove second derivative test.
-Proposition 4.3.3.
-<li>Add exercises 4.3.9, 4.3.10, 4.3.11.
-<li>Rewrite proof of Lemma 4.4.1, and use clearer variable names.
-<li>Add figure to Example 4.4.5.
-<li><b>Modify Exercise 4.4.6 very slightly</b>, replace ``interval'' with
-``open interval''.  The distinction is irrelevant for how one proves it
-and considering other types of intervals makes the proof longer.
-<li>Add figure to the proof of Proposition 5.1.7.
-<li>Add figure to Proposition 5.1.10.
-<li>Add figure to Example 5.1.12.
-<li>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.
-<li>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.
-<li>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.
-<li>Add exercise 5.2.17.
-<li>Add figure to proof of the fundamental theorem of calculus in 5.3.
-<li>Add remarks about other definitions of logarithm and the exponential,
-and about the uniqueness and existence following from a subset
-of the given conditions.
-<li>Add exercise 5.4.11.
-<li>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.
-<li>Add figure for integral test for series in 5.5.
-<li>Add figure to example 6.1.4.
-<li>Add figure to definition uniform convergence in 6.1.
-<li>Add exercise 6.1.12, 6.1.13, 6.1.14.
-<li>Add subsection to 6.2 on swapping of limit of functions and derivatives
-for continuously differentiable functions.
-<b>This makes exercise 6.2.7 much easier</b> as we essentially do the main
-bit as a theorem.  There is a new figure in this subsection.
-<li>Add subsection to 6.2 on convergence, differentiation, and integration
-of power series.
-<li>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.
-<li>Add exercises 6.2.15, 6.2.16, 6.2.17, 6.2.18, 6.2.19, 6.2.20, 6.2.21.
-<li>Add remark about weaker solutions to ODEs using the integral equation.
-<li>Use the more common interior notation in 6.3, and in 7.6.
-<li>Add exercises 6.3.7, 6.3.8, 6.3.9.
-<li>Improve triangle inequality figure in 7.1.
-<li>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.
-<li>Add exercises 7.1.9, 7.1.10, 7.1.11, 7.1.12, 7.1.13.
-<li>Improve the open set figure in 7.2.
-<li>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.
-<li>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.
-<li><b>Replace exercise 7.2.5</b>, the conclusion was already proved
-in exercise 1.4.3 (in more generality, in fact).
-<li><b>In Exercise 7.2.12</b> the implication goes the other way (erratum in
-earlier versions), as is needed in the text.
-<li>Add figures to Propositions 7.2.9, 7.2.13, 7.2.15, and 7.2.26.
-<li>Add $(0,\infty)$ and $[0,\infty)$ as an examples of an open and
-closed sets in ${\mathbb{R}}$ to Example 7.2.5.
-<li>Add footnote about empty sets and connectedness.
-<li>Add exercises 7.2.15, 7.2.16, 7.2.17, 7.2.18.
-<li>Add figure to definition of convergence in 7.3.
-<li>Add example to 7.3 of $C([0,1],{\mathbb{R}})$ where convergence is the
-uniform convergence.  This renumbers the following examples, propositions,
-etc...
-<li>Add remark that pointwise convergence does not come from a metric.
-<li>Add example for convergence in the complex numbers.
-<li>Add exercises 7.3.13, 7.3.14.
-<li>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...
-<li>Add proposition that $C([a,b],\R)$ is a complete metric space.
-<li>Add proposition that a closed subset of a complete metric space is
-complete, that is used later.
-<li>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.
-<li>Add an example for the Lebesgue covering lemma, finding a $\delta$ for a
-cover.
-<li>Add figures to proof of Proposition 7.4.9, Lebesgue covering lemma, and 
-Theorem 7.4.11.
-<li>Add exercises 7.4.17, 7.4.18, 7.4.19, 7.4.20.
-<li>Add figure for Lemma 7.5.7.
-<li>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.
-<li>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.
-<li>Make notation more in line with the rest of the chapter in 7.6.
-<li>Move all exercises to the Exercises subsection 7.6 to be consistent
-with the rest of the book.
-<li>Add exercise 7.6.11.
-</ol>
+No new changes since 5.0