%\section{\defs} \textwidth=7in \evensidemargin=-.25in \oddsidemargin=-.25in \newtheorem{thm}{Theorem} %[section] \newtheorem{cor}[thm]{Corollary} \newtheorem{prop}[thm]{Proposition} \newtheorem{lem}[thm]{Lemma} \newtheorem{ex}[thm]{Exercise} \newtheorem{pres}{Presentation} \theoremstyle{definition} \newtheorem{defn}[thm]{Definition} \newtheorem{notn}[thm]{Notation} \newtheorem{eg}[thm]{Example} \newcommand{\R}{{\Bbb R}} \newcommand{\GL}{\operatorname{GL}} \begin{document} \title{Analysis Seminar: Assignment 7} \maketitle \section*{Reading} \begin{enumerate} \item Read section 23. As used on page 201 the word ``discrete'' has the following meaning. \begin{defn} A subset \( S \) of a metric space is discrete if it has no cluster points. \end{defn} You should note that this is stronger than saying that \( S \) consists entirely of isolated points. \item Read section 24. \item Read section 25. \end{enumerate} \section*{Doing} In each case I list what is a reasonable prioritization. Your own priorities may differ a bit from mine. \begin{enumerate} \item All the problems at the end of section 23 are worth doing. If you are not confused by the reading, I would prioritize them in order: 3, 6, 5, 4, 2, 1. If you are confused by the reading, I would prioritize them in order: 1, 2, 3, 4, 5, 6. Do at least two or three of your high priority problems. \item Problems 3--6 at the end of section 24 are all applications of the theorem in problem 2. You may wish to do these before proving the result in 2. If you are comfortable with the material in this section I prioritize the problems: 2, 5, 6, 3, 4, 1. If not, 1, 3, 5, 4, 6, 2. Again, do at least two or three of your high priority problems. \item The problems at the end of section 25 are mostly interesting, but only 4 and 8 involve the real material of the chapter more than other sorts of cleverness. I would prioritize them as follows: 8, 4, 3, 1, 2, 5, 6, 7. Do at least one or two of your high priority problems. \end{enumerate} \section*{Writing} Expertly write up at least one page of work you are proud of from this week. Hand this in no later than than noon on Tuesday following Break. \section*{Presentations} \begin{pres}[Tom] The definition of manifold and an example or two. \end{pres} \begin{pres}[Rob] Theorem 24.4 and either solution or confusion involving problem 24.2 \begin{pres}[Mike] The definition of integral and the lemmas involved in showing that the integral is well-defined. \end{pres} \end{document}