diff --git a/changes-draft.html b/changes-draft.html
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--- a/changes-draft.html
+++ b/changes-draft.html
@@ -1,109 +1,3 @@
This file is a draft of the new changes for http://www.jirka.org/ra/changes.html
-The theme of this revision is trying to fix all the minor issues and errata I
-could find, and improve clarity, but not make any large changes, nor add any
-content. Perhaps the biggest change are 3 new exercises. Two are there to
-fix an erratum, and one was sort of kind of hidden part of an old exercise.
-There are a few new explanatory sentences here and there,
-but nothing new beyond that.
-
-Add a very short finite example of cartesian product after
- Definition 0.3.10.
-Make Definition 0.3.11 (function) a bit easier to read by explicitly
- stating that it is the $y$ that is unique.
-In Definition 0.3.11 (function), define codomain. It does appear in at
- least one place in the book, and it may be good for the sake of being a
- reference book.
-Example 0.3.32 is a bit too informal and just leaves out 0 and the
- negatives, so add that.
-Move the argument for why infimum and supremum are unique to right
- after Definition 1.1.2 and note why this means that the notation is
- well-defined.
-In Definition 1.1.1 (ordered set), label "transitivity" and "trichotomy"
-Remove the first sentence of the proof of Proposition 1.1.9 and just
- give the example before the proof. It is not really part of the "proof" of
- the statement itself.
-In the proof of Example 1.2.3, the second displayed estimate, the $h$ is
- given as an equality, so the last $\leq$ is actually $=$.
- Also show explicitly that $s-h > 0$ to fix erratum.
-Improve the wording of proof of 1.4.2, also in the same proof the sets
- $A$ and $B$ were being defined but we only used $=$ and not $:=$. Also,
- change $b_k$ to just be any number in $(a_k,b_{k-1})$, that is simpler and
- sufficient.
-In Definition 2.1.9, move the "Some authors use the word monotonic." to a
- footnote to simplify the definition.
-After definition 2.1.9 mention $\{ n \}$ as an example of a monotone
- increasing sequence.
-Simplify the proof of Proposition 2.1.10. Don't say anything about the
- $B$, we never use the bound, just say the set of values is bounded, that is
- good enough to compute the supremum.
-After Definition 2.1.16, explicitly mention what we mean by a subsequence
- by writing $x_{n_1},x_{n_2},x_{n_3},\ldots$.
-Add Exercise 2.1.23
-In Proposition 2.2.11, recast the proof of unboundedness to not be a
- contradiction proof. It's the same idea, but it avoids having to explain
- why it is a contradiction, and avoids a contradiction proof.
-In Example 2.2.14, use $M$ instead of $N$ for consistency.
-Add Exercise 2.3.20
-Before Proposition 2.5.6, make "tail of a series" a defined term and add
- it to index.
-Rephrase the last argument in the proof of Proposition 2.5.17 to be a
- little bit more straightforward.
-Add useful remark to Exercise 2.5.6.
-Add remark to Exercise 2.5.16 about starting the series, and that only
- tails satisfy the hypotheses, so that students do not forget to check these
- technicalities.
-Add footnote on $L=\infty$ to proof of Proposition 2.6.1
-Add a better introductory sentence to cluster points in 3.1.1.
-In Lemma 3.1.7 and Proposition 3.1.17, add $L \in {\mathbb{R}}$ to the
- hypotheses, that makes it clearer that it is a given number.
-Since every semester I get a question about Exercise 3.1.1, add a
- parenthetical remark: Yes one must prove the limit is what one claims
- it is.
-In Exercise 3.1.11, change "Then show $f(x) \to L$ as $x \to c$ for some
- $L \in {\mathbb{R}}$" to "Then show that the limit of $f(x)$ as $x \to c$ exists."
- Perhaps that will make students not start on the wrong path of starting with
- some $L$ existing rather than proving that it exists.
-When proving the Thomae function (3.2.12) is continuous at irrational
- numbers, note that since the limit of $\{ x_n \}$ is $c$, then every
- rational number is in the sequence at most finitely many times.
-At the end of example 3.2.13, mention that $g$ is in fact continuous on $B$.
-After proof of Lemma 3.3.1, add a short paragraph highlighting the use of
- Bolzano-Weierstrass, to emphasize the technique. It changes the pagination
- of 3.3 a tiny bit (inadvertently getting less jarring page breaks)
-Reword slightly the end of the proof of Example 3.4.3 to improve clarity.
-Add two lines of text after proof of Theorem 3.4.4 to make a similar
- point as for 3.3.1, again changing pagination of the rest of 3.4 very
- slightly.
-The "In other words" of Exercise 4.1.14 is confusingly stated with an
- inequality, while the way to prove it is simply with an equality, that was
- a cut and paste typo. Of course it is true with an inequality still.
-In Exercise 4.1.15 (simple L'Hospital's rule) note that the limit of the
- quotient of derivatives must exist, no need to "suppose" it, we're assuming
- here that the derivatives are continuous and the denominator is never
- zero. Also assume that $g(x)\not= 0$ if $x \not= c$. While it can be
- proved that $g(x) \not= 0$ in some neighborhood of $c$, that was not
- intended in this simple version.
-Added Exercise 4.1.16 to keep this sort of exercise explicitly. That
- is, if $f'(c) > 0$, then show that $f(x)$ is negative a bit before $c$
- and positive for a bit after $c$, thus zero only at $x=c$.
-Be a little bit more precise in the proof of Lemma 4.2.2 to say that
- all the $x$ and the $y$ are still within $\delta$ of $c$.
-In the proof of Proposition 4.2.6 (and also 4.2.7) note explicitly
- that $[x,y] \subset I$ because $I$ is an interval.
-In Exercise 4.2.9, add a note that the student needs to prove that
- $g(x)$ is not zero for $x \not= c$ so that the left hand side of the
- equality makes any sense at all.
-In Exercise 4.3.2, ask about the $d$th Taylor polynomial, not the
- $(d+1)$th, that was a typo. Though of course the exercise is still true
- for $d+1$.
-In Definition 5.1.6 and proof of Proposition 5.1.7 use $\ell$ instead of
- $m$ since $m$ is used all over the place for a minimum of the function.
-In proof of 5.3.5, explicitly mention the domain of $F$ for clarity.
-In Propositions 7.2.6 and 7.2.8 explicitly mention that the sets are
- subsets of $X$.
-Throughout get rid of the use of the word "any" where it could be ambiguous.
-Fix a couple of uses of "=" where ":=" is more appropriate.
-Improve the typesetting of some statements.
-Some minor clarifications and tightening of the language a bit throughout
- the book.
+
diff --git a/changes2-draft.html b/changes2-draft.html
index 1caf708..6331fa7 100644
--- a/changes2-draft.html
+++ b/changes2-draft.html
@@ -1,8 +1,3 @@
This file is a draft of the new changes for http://www.jirka.org/ra/changes2.html
-Make line in Figure 9.6 a bit bolder to make it easier to pick out.
-In Proposition 10.3.7, use $\ell$ for the number of balls to make it
- clear that the number is quite likely different from the number of
- rectangles.
-Fix a couple of uses of "=" where ":=" is more appropriate.
-Some minor clarifications and fixes to style and grammar.
+
diff --git a/extra.css b/extra.css
index e9ce549..edef4ef 100644
--- a/extra.css
+++ b/extra.css
@@ -73,6 +73,6 @@ li { display: list-item; }
}
hr.rahr {
border-style: none;
- background-color: #666;
- height: 1px;
+ height: 0px;
+ border-top: thin solid #666;
}