A couple of minor clarifications

TODO

@@ -4,6 +4,30 @@ HTML/PreTeXt:
 
 Possibilities in the long term (possibly next possibly other edition):
 
+* Vol I: Add another example for computing limits using the theorems to
+  subsection 2.2.2?  Perhaps not needed, but might be good.
+  (thx Manuele Santoprete)
+
+* Vol I: Perhaps be always a bit more explicit with the domain on both
+  "uniformly continuous" and "Lipschitz"
+  (thx Manuele Santoprete)
+
+* Vol I: A bit more about Dedekind vs Cauchy completeness?
+  (thx Manuele Santoprete)
+
+* Vol I/II: Add more names to the theorems/propositions.  E.g.
+  From Manuele Santoprete:
+  "I think it is good to give names to theorems, since it make it easier to
+   communicate math. There are several spots where one can give a name to the
+   theorems. The most important one is Proposition 2.1.10. This is often called
+   the "Monotone Convergence Theorem". I think at least this name should be
+   used.  Also the min-max theorem is often called extreme value theorem. I
+   think it makes sense to mention this, since many students are familiar with
+   the term from calculus.  Bartle's book has some very nice names. Another one
+   is the Cauchy Convergence Criterion (Theorem 2.4.5). There are others as
+   well, but I did not make a note of them. Moreover,  Lemma 2.2.3 sometimes is
+   called the Comparison Theorem."
+
 * Vol I: Perhaps avoid the whole "well ordering of N" and "induction"
   equivalence bit.  We could just assume induction as axiom and have well
   ordering as a proposition.  Might require replacing the exercise 0.3.18

ch-contfunc.tex

@@ -1870,7 +1870,11 @@ \subsection{Uniform continuity}
 it only depends on $\epsilon$.  The domain of definition
 of the function makes a difference now.  A function that is not uniformly
 continuous on a larger set, may be uniformly continuous when restricted to a
-smaller set.  Note that $x$ and $c$ are not treated any differently
+smaller set.
+We will say \emph{uniformly continuous on $X$} to mean that
+$f$ restricted to $X$ is uniformly continuous, or perhaps to just emphasize
+the domain.
+Note that $x$ and $c$ are not treated any differently
 in this definition.
 
 \begin{example}

ch-real-nums.tex

@@ -546,6 +546,7 @@ \subsection{The set of real numbers}
 real number $c$ such that
 $a < c < b$.  Take, for example, $c = \frac{a+b}{2}$ (why?).  In fact,
 there are infinitely many real numbers between $a$ and $b$.
+We will use this fact in the next example.
 
 The most useful property of $\R$ for analysts
 is not just that it is an ordered field, but that it has the

ch-vol1-intro.tex

@@ -120,7 +120,8 @@ \section{About this book}
 Jim Brandt, Kenji Kozai, Arthur Busch,  Anton Petrunin,
 Mark Meilstrup, Harold P.\ Boas, Atilla Y{\i}lmaz,
 Thomas Mahoney, Scott Armstrong, and Paul Sacks,
-Matthias Weber, Robert Niemeyer, Amanullah Nabavi,
+Matthias Weber, Manuele Santoprete,
+Robert Niemeyer, Amanullah Nabavi,
 for teaching with the book and giving me lots of useful feedback.
 Frank Beatrous wrote the University of Pittsburgh version extensions, which
 served as inspiration for many more recent additions.

changes-draft.html

@@ -1,37 +1,3 @@
 This file is a draft of the new changes for http://www.jirka.org/ra/changes.html
 
-<li>When stating that well ordering of $\mathbb{N}$ and induction are
-  equivalent, hedge our bets with "in a sense" and add a footnote mentioning
-  that we are really assuming $n-1$ exists (which is obvious from the proof).
-  In a related change, make exercise 0.3.18 just straight to the point and
-  don't mention the equivalence.
-<li>Reword the beginning of the proof of Example 1.2.3.
-<li>Make Proposition 1.4.1 more readable.
-<li>In the proofs of Propositions 2.3.2, 2.3.6, Example 3.1.8, when referring to a sequence, always
-  use braces.
-<li>Add explicit link/reference to chapter 7 to remark 2.4.6.
-<li>Add another example restriction on which $g$ is continuous to Example 3.2.13.
-<li>Reword Lemma 3.3.1 and Theorem 3.3.2 (min-max) as a simpler single sentence.
-<li>Make the Min-max theorem alternatively titled "Extreme value theorem" which
-  is more common.  Emphasize the "closed and bounded" in the theorem
-  statement.
-<li>Reword the proof of Proposition 4.1.10 a little bit to make it clearer.
-  Also include a tiny bit more motivation.
-<li>Add some intuition (being in a fog analogy) to the intro for critical points.
-<li>Add Remark 5.1.15 to say something about integral being a sum and being
-  global as opposed to derivatives being local.
-<li>In 5.2 add a few more motivating sentences.  And merge the paragraph from
-  below Proposition 5.2.5 to the one above it to make things flow a little better.
-<li>Mark 5.5.13 as "Integral test," and add that to the index.
-<li>In 6.3, define "initial condition" as a term.
-<li>At the end of 6.3, reword the quip about continuity being necessary and add a
-  very short remark about Peano existence if $F$ is only
-  continuous, to justify Example 6.3.6 being discontinuous.
-<li>Name the identity bit of Definition 7.1.1 the "identity of
-  indiscernibles."  It is a bit wordy, but it feels like the property ought to
-  have a name if the others do.
-<li>Reword proof of Proposition 7.2.14 to do the forward direction first.
-<li>Add $A \subset \overline{A}$ to Proposition 7.2.19 rather than just in text.
-  The proof flows much nicer then.
-<li>Several small clarifications.
-<li>A few more explicit references/links.
+

changes2-draft.html

@@ -1,33 +1,2 @@
 This file is a draft of the new changes for http://www.jirka.org/ra/changes2.html
 
-<li>On page 11, add a short note about the $d+1$ linearly independent
-  vectors from the definition of dimension.
-<li>In Exercise 8.4.7, assume $q$ is not identically zero.  The result is
-  vacuously true even if $q$ is identically zero, but there is no reason to
-  make students think about this rather stupid technicality.
-<li>The proof in Example 8.1.25 is hopefully clearer.
-<li>In definition 8.3.8, remove the definition of the notation $D_j f$.  We
-  never used it later.
-<li>In Definition 9.2.1, change the definition of "simple" for non-closed
-  paths.  Typically a path that bites itself back in the middle is not called
-  simple, so rule out that case.
-<li>Rename subsection 10.3.3 to "Images of null sets under differentiable
-  functions".
-<li>In proof of Lemma 10.3.9, say how to prove it for open balls.
-<li>Improve definition 10.6.1 to be (much) simpler (though equivalent): Only
-  take a finite disjoint union of simple closed sets as we are assuming $U$
-  is bounded anyway.
-<li>Reorder the proof of Theorem 10.7.2 a little bit to make it more logical.
-<li>On page 148, add a short note that the $e^{z+w}=e^ze^w$ leads to a quick
-  computation of the power series at any point.
-<li>In Corollary 11.3.7, emphasize that that $a$ is any complex number,
-  since just above it was a real number.
-<li>In definition of the exponential on page 147 (the definition of
-  $E(z)$) explicitly say that this means that it is analytic.
-<li>Be more precise in Exercise 11.4.9 to say to derive the power series at the
-  origin.
-<li>In Example 11.6.3, emphasize that $f_n$ are continuous.
-<li>In 11.8.2, when saying we could develop everything with sines and
-  cosines, give the actual form of the series and refer to Euler's formula,
-  so that when we later call such series also Fourier series, the reader is
-  not confused.