From fd75aa9266e30f5d463d8629aa6581dd7a7644f6 Mon Sep 17 00:00:00 2001 From: "Jiri (George) Lebl" Date: Tue, 9 Nov 2021 13:40:18 -0600 Subject: [PATCH] Fix up css for the rahr and move changes to site --- changes-draft.html | 108 +------------------------------------------- changes2-draft.html | 7 +-- extra.css | 4 +- 3 files changed, 4 insertions(+), 115 deletions(-) diff --git a/changes-draft.html b/changes-draft.html index 135d8b8..1a4ab2b 100644 --- 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; }