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This file is a draft of the new changes for https://www.jirka.org/ra/changes.html

General changes:

<li>In definitions of limits (sequence, continuous limits, in metric spaces,
etc), don't "cheat" and say "if a limit is unique." While it feels a
little wordy since the first thing we do is prove that the limit is
unique, I'm starting to feel that this may be contributing to confusion
about proof writing to students.
<li>Do not shorten sequences to \(\{ x_n \},\) but always write out
\(\{x_n\}_{n=1}^\infty.\) It seems to me this shorthand is
causing more confusion than it is worth, especially with
regards to the distinction between set and sequence, and also
when working with subsequences.
<li>Do not shorten sequence limits to \(\lim\, x_n,\) but always write out
\(\lim_{n\to\infty} x_n.\) It seems to me this shorthand is
causing more confusion than it is worth, especially
when subsequences are introduced.
<li>Do not shorten series to \(\sum\, x_n,\) but always write out
\(\sum{n=1}^{\infty} x_n.\) It seems that it is making students
forget a limit is involved.
<li>Uniform norm notation is changed to \(\|\cdot\|_K,\)
where \(K\) is the set where the supremum is taken. I've had a number of
complaints about the \(u\) notation not being very standard. And this goes
better with my other books where I use the more standard notation.
<li>Add parentheses to the notation for "Riemann integrable" for consistency,
that is, \(\mathscr{R}([a,b])\) instead of \(\mathscr{R}[a,b] .\)

Specific larger changes:

<li>In 0.3, after definition of composition state as an "exercise"
that compositions of bijections are bijections. This is actually a
WebWorK exercise.
<li>Add Exercises 0.3.26 and 0.3.27 to prove the DeMorgan's laws and
the pushforward/pullback propositions for infinite unions and
intersections.
<li>Add Figure 1.3 on the set \(\{ \frac{1}{n} : n \in {\mathbb{N}} \}\)
and its infimum (Corollary 1.2.5) (renumbers all the following figures in
chapter 1).
<li>Simplify proof of Proposition 2.1.7 by just defining the \(B\) once
rather than defining two different bounds.
<li>As per suggestions name 2.1.10 the "Monotone convergence theorem"
and therefore make it a Theorem rather than a Proposition.
<li><b>Exercise 2.2.9, add hypothesis that \(x_n\not=x\) for all \(n.\)</b>
It is implied in that the limit makes sense,
but it should be stated explicitly.
<li>Replace Example 2.4.3 with a more useful example, one that isn't Cauchy.
Suggested by Harold Boas.
<li>Clean up the proof of Proposition 2.6.2, the Alternating series test.
Mainly improve the readability by using the variable names more
consistently, rewrite the end of the proof, and fix an erratum.
<li>Add Figure 2.8 showing graphically why the alternating sum converges.
<li>Add Figure 2.9 to show how the sample rearrangement of the alternating
harmonic sum converging to 1.2 works. Renumbers the following figure
chapter 2.
<li>Add Exercise 2.6.15 for Tonelli/Fubini for sums. This is too useful of
a "variant" of reordering not to have it, plus we do use it in
volume II in a proposition.
<li>Add Figure 3.1 for Example 3.1.6 where limit is different from
value (renumbers all figures in chapter 3).
<li>In Corollaries 3.1.9, 3.1.10, 3.1.11, the hypothesis is only needed
for all \(x \in S \setminus \{ c \},\) as we do in all the other
results of this section. The way it is stated could
be confusing, so change them to this hypothesis. (It is equivalent
because one can always replace \(S\) with \(S \setminus \{ c \} \)
of course. <b>This affects
Exercises 3.1.3 and 3.1.4, but it at worst makes them slightly less
confusing and more straight forward.</b>
<li><b>Replace Exercise 3.1.10</b>. This exercise was almost exactly the
same as 3.1.11.
<li>Rename section 3.3 to "Extreme and intermediate value theorems".
<li>Add a new Example 3.3.11 showing the existence of roots,
so the old 3.3.11 becomes 3.3.12, and
Corollary 3.3.12 becomes 3.3.13. This is a nice application,
and ties in some prior results, and it is good to see it especially
if 4.4 is not covered.
<li>Add Figure 3.8 for the Corollary 3.3.13 (was 3.3.12) where
image of a continuous function is an interval.
<li>Add Figure 3.10 to visualize why the square root is not Lipschitz.
<li>Make the Lemma 4.2.2 be stated for an open interval \((a,b)\) since we
don't need the endpoints and it could really just be confusing.
<li>Add Exercise 5.1.15
<li>In the proof of Proposition 5.2.2 mention boundedness to be completely
rigorous.
<li>Add Exercise 5.2.18.
<li>In Exercise 5.3.7, add \(a+\epsilon \lt b-\epsilon\) to emphasize where
things are well defined.
<li>Add Exercise 5.3.13.
<li>Add Figure 5.6 and Figure 5.7 giving the logarithm and the exponential,
the later figures in chapter 5 are renumbered.
<li>Add Figure 6.7 to Example 6.2.9 to illustrate what is happening.
<li>Add Exercise 6.2.22.
<li>Add Figure 6.8 in subsection 6.3.1 to demonstrate a first order ODE
as a slope field.
<li>In Example 6.3.3, refer back to the figure for the exponential which
shows the slope field. Also add figure showing the exponential together
with the first few iterates.
<li>In Remark 6.3.7, use \(x\) instead of \(t\) for the Heaviside function
to make things less confusing.
<li>Add Figure 7.2, to clarify especially the end of Example 7.1.3. This
renumbers the rest of figures in chapter 7.
<li>Add \(\{x\}\) as an example of a closed set to Example 7.2.5, but leave
the proof to the online homework (it is rather simple).
<li>To be more consistent, and avoid overuse of the letter x for everything,
use \(p\) instead of \(x\) in Propositions 7.3.11, 7.3.12, 7.3.13, 7.4.2,
Exercises 7.3.1, 7.3.5, 7.3.7, and Definition 7.4.2
<li>Move the remark about subspaces not being complete after Proposition
7.4.5 as it makes more sense that way.
<li>In Definition 7.4.7, Example 7.4.8, and the proof of Proposition 7.4.9,
use \(m\) instead of \(k\) to avoid overusing the letter \(k,\) to make
it easier to talk about the proof.
<li>Add a paragraph after Proposition 7.4.9 emphasizing the difference
between compactness and closedness in the sense that "compact" doesn't
care about the ambient topology while "closed" most definitely does.
<li>Rewrite the \(n=1\) part of the proof of 7.4.13 (Heine-Borel) to be a
little bit more like \(n=2\) part (use the proposition to reduce to
closed and bounded interval) and just refer to Example 2.3.8 instead of
repeating the argument.
<li>Change the mapping in Theorem 7.6.2 (Contraction mapping principle)
to \(\varphi\) from \(f\) to avoid overusing \(f\) in this section.
<li>Mention that \(d\) is the uniform norm on \([-h,h]\) in the proof.

Smaller changes:

<li>Many minor rephrasings and rewordings for added clarity.
<li>In the definition of \(S\) in the proof of Theorem 0.3.6,
use \(n,\) to avoid overloading of \(m.\)
<li>Say that \(A\) is a set in Cantor to be a bit more precise.
<li>Simplify parts (iii) and (iv) of Definition 1.1.2.
<li>In proof of part (v) of 1.1.8, should just use the definition rather than
part (ii) of the proposition.
<li>Improve wording around Proposition 1.2.2, remove some unnecessary words
and explicitly state the version with \(|x|,\) which is a common
statement.
<li>In the proof of 1.2.6, use \(c\) instead of \(b\) for the second
inequality to avoid overloading \(b.\)
<li>Improve slightly the wording in the examples after definition of a limit
of a sequence.
<li>After Proposition 2.1.10, make a remark about monotone sequences and
boundedness above/below.
<li>Improve wording of Example 2.1.12.
<li>After definition of subsequences, give a little bit more detail of the
example subsequence.
<li>Improve the recursive sequence (Newton's method) wording slightly.
<li>In the proof of Proposition 2.2.11, use \(x\) instead of \(L\) in the
proof as \(L\) is used in the related Lemma 2.2.12 for something else.
<li>In the proof of Theorem 2.3.4, when defining the subsequence, suppose
\(n_1,\ldots,n_{k-1}\) is defined and define \(n_k\). That way it
is more consistent with the rest of the proof and should be easier
to follow. Also say \(m \geq n_{k-1}+1\) instead of \(m \gt n_{k-1}\)
to make it clearer where the \(+1\) comes from.
<li>Change the index variable in Proof of Proposition 2.3.6 from \(j\) to
\(n\) for consistency.
<li>Simplify the remark after Definition 2.3.12 as it may be hard to parse.
<li>Be more precise with the hint and the indexing in Exercise 2.3.7. Also
mention that it is just one of the possible proofs (I find it a cool proof).
<li>Remark 2.4.6 should refer to theorem not proposition. Also clarify that
Cauchy completeness means that the limit should be back in the set.
The remark is purposefully vague (to omit the gory details),
it's not really a definition, nor a construction of the reals,
but we don't want to be misleading.
<li>In Definition 2.5.1 don't define an extra variable \(x\) just for the
limit.
<li>In Definition 2.5.14 be consistent with wording for absolute
and conditional convergence. That is, change "is conditionally
convergent" to "converges conditionally", and add both "converges
conditionally" and "conditional convergence" to the index.
<li>In 2.5, when talking about the terms going to zero "fast enough"
before the comparison test, this is about series with positive terms,
so make that clear.
<li>Throughout, where appropriate, use \(i\) or another letter instead of \(j\)
as that typesets a lot better with series and as powers.
In some places this also changes the other indices.
<li>Rephrase Merten's theorem a tiny bit.
<li>In Exercise 2.6.11 part c, be more precise in the parenthetical remark
about divergence.
<li>Reword Exercise 2.6.4 part a) to be a little easier to understand
what is being asked
<li>Before Proposition 3.1.15, emphasize the meaning of it, that it means
that the limit is "local."
<li>Below Definition 3.2.1, we make a statement about the converse not
holding, but with no reference. An example is given in Example
3.2.13 so give a parenthetical reference to it.
<li>While fixing the labels in Figure 3.4 (was 3.3), make them smaller so that they
don't run into each other and move them below the axis.
<li>Improve wording of Example 3.2.12.
<li>Before Lemma 3.3.1, emphasize that \([a,b]\) is a closed and bounded
interval. In a related change, this was emphasized in the statement
of the Min-Max/Extreme value theorem, but that was making it too
wordy, so make a remark right after the theorem and simplify the
statement.
<li>When defining absolute minimum/absolute maximum, say that these are
what \(f(c)\) is (as is shown in the figure).
<li>Improve the wording of the examples 3.3.4, 3.3.5, 3.3.6 to emphasize
which properties are satisfied and which are not.
<li>At the end of the proof of Proposition 3.3.10, when we claim the root
by Bolzano, say it is in the open interval so that the claim lines up
better with the theorem.
<li>Remove the "definition" of the phrase "uniformly continuous on \(X\)"
as we only ever use it in a remark and it is not standard verbiage anyway.
<li>Rewrite the introductory paragraph to section 3.4.2 a bit, and
make the statement of Lemma 3.4.5 slightly more precise.
<li>Clean up Figure 3.9 very slightly.
<li>In Example 3.4.10, name the second function \(g\) to make things
hopefully a bit clearer.
<li>Reword Example 3.5.3 a little.
<li>Make caption to Figure 3.12 a bit more precise.
<li>Clean up Figure 4.1 very slightly.
<li>In Lemma 4.2.2, move the initial sentence of the proof to the end and
reword it.
<li>After Theorem 4.2.4, say explicitly that the slope of the secant line
is the mean value of the derivative, hence the name of the theorem.
<li>In subsection 4.2.4 (applications of the mean value theorem) add a short
description of how the applications work: by getting rid of a limit.
<li>At the end of Example 4.2.12, be a little less wordy.
<li>Reword the paragraph in front of Corollary 4.4.3, now that we have the
existence of roots as an explicit example in 3.3.
<li>In Exercise 4.4.2, remark that it is the same as Exercise 4.1.10, to avoid
possibly assigning this type of problem twice.
<li>In Proof of Proposition 5.1.7, use (in addition to changing \(j\) to
\(i\)) \(q\) instead of \(p\) since I just realized that this is a
terrible name since the partition is \(P.\) Also changes Figure 5.2.
<li>Before Proposition 5.1.13, refer to Figure 5.1 for intuition of what
the difference of the upper and lower sums measures.
<li>Explicitly mention in the beginning of the proof of Lemma 5.2.7
that \(f\) is bounded.
<li>Mention in a footnote that people often say "converges" when they mean "converges
pointwise"
<li>Improve the wording of Exercise 6.1.10.
<li>Reorganize the setup in proof of Proposition 7.2.11 a bit.
<li>Add a small note after Heine--Borel (7.4.14) to emphasize it does not
hold in subspaces of \({\mathbb{R}}^n.\)
<li>In Exercise 7.5.2, reference the figure with the graph.
<li>Rewrite Proposition 7.4.9 to emphasize in which direction the
implication goes.
<li>Use \(z\) instead of \(\tilde{y}\) in the proof of Proposition 7.5.12
<li>In 6.3 and 7.6, mention for completeness that
\([h-x_0,x_0+h] \subset I\) in the statement of Picard's theorem.
<li>In 7.6, improve the wording of the proof of Picard's theorem.
<li>Improve the wording in part c) of Exercise 7.6.9. The point of applying
the theorem is not to find that \(\sqrt{2}\) is a fixed point, that follows
just from the formula. The point is that the theorem does apply.
Add a note about why this is useful.
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