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\begin{document}
\title{Math 101--Analysis Seminar}
\maketitle
\section*{A preliminary note}
Dear Analysts,
\bigskip

In a previous message I wrote:

\begin{quotation}
Some time before next semester begins we have to prepare for our first
meeting.  (Otherwise we won't have enough to talk about for a full
meeting.) I will send out a more formal assignment during break, but
roughly speaking we will start by discussing Chapter 1 in the text.
(Analysis on Manifolds by Munkres) This should be mostly review for
most of you.  Look at this chapter and the associated problems over
break.  If you find anything which looks totally alien, let me know if
you can so I can adjust my expectations.
\end{quotation}

Here, I would like to make some more specific suggestions.

I understand
that not all of you will be able to do all of this, but if all of you do
what you can, we will have enough to talk about for our first meeting.   I
will hand out a more formal syllabus at our first meeting.

Throughout, I will make a distinction between doing'' problems and writing''
problems.  I expect that you will write down some sort of
solution for your own records when I ask you to do a problem.  I don't
expect this record to be necessarily expertly written, but I do
expect it to be complete in the sense that little or no mathematical
thought need be added in the process of writing up the solution
you to write a problem, I will want a polished and careful writing

Read sections one through four in Munkres.
\section*{Doing}
As you read Section 1 be sure to ask yourself whether you have some
idea of how to prove each of the unproved assertions.  Assertion 1.3 will
occur as exercise 1.2.
\begin{ex}
Do exercises 1.1 through 1.4.  Look at exercise
1.5 and do it if you have time.
\end{ex}
\begin{ex}
Look at all the problems in section 2.  Do the first one and the last
one and any of the others which don't look familiar.
\end{ex}
\begin{ex}
Show that a square matrix with integer entries is invertible if and
only if it has determinant $$\pm 1$$.
\end{ex}
\begin{ex}
Look at all the problems in section 3.  Do 3.6
\end{ex}
\begin{ex}
In section 4 (and afterwards) Munkres occasionally uses the sup norm
and occasionally uses the Euclidean norm.  In section 4, find all
instances of each and try to explain to yourself why he uses which
norm.
\end{ex}
\begin{ex}
Do problems 4.1, 4.2, 4.3, and 4.4.
\end{ex}
\section*{Writing}
Write up one or more of the problems you have done and hand it in to me so
that I can critique the result.  Altogether, I would like a page or two of
carefully written exposition.  If possible, pick from among the problems
that you did not know how to do before you did them.
\end{document}