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\textsf{\textbf{{\Large Mathematics 424B/574B}}}
\hfill \textsf{\textbf{{\Large Autumn 2007}}}

\textsf{\textbf{Study problems for the final}}

There will be some problems on the final which are based on some of
these; there will also be some problems on the final which are not
based on any of these.

\begin{center}
\underline{\hspace{3in}}
\end{center}

\begin{enumerate}
\item 
Exercises 3.6, 3.12, 3.18
\item 
Let $\{a_{n}\}$ be a sequence of real numbers, let $\{p_{n}\}$ be a
sequence of positive integers, and let
\[
c_{n} = \frac{p_{1} a_{1} + p_{2}a_{2} + \dots + p_{n}a_{n}} 
{p_{1}+p_{2}+\dots +p_{n}}.
\]
Prove that if the sequence $\{a_{n}\}$ converges to some number $a$,
then $\{b_{n}\}$ converges to $a$ also.
\item 
Let $\sum_{n=0}^{\infty} a_{n}$ be a series.  For each $n$, let
$s_{n}=a_{0}+\dots + a_{n}$.  Let $\{b_{n}\}$ be a sequence.  Suppose
that 
\begin{enumerate}
\item [(i)] the sequence $\{s_{n}b_{n+1} \}$ converges, and
\item [(ii)] the series $\sum_{n=0}^{\infty} s_{n}(b_{n}-b_{n+1})$ converges.
\end{enumerate}
Prove that the series $\sum_{n=0}^{\infty}a_{n} b_{n}$ converges.
(Theorem 3.41 might be helpful.)  Use this to give another proof of
Theorem 3.42, and also use it to solve exercise 3.8.
\end{enumerate}

\begin{center}
\underline{\hspace{3in}}
\end{center}

\begin{enumerate}
\item
Exercise 4.4
\item 
Given a function $f: X \rightarrow Y$, prove that $f$ is continuous if
and only if $f(\overline{E}) \subseteq \overline{f(E)}$ for every
subset $E$ of $X$.
\item
Suppose that $f:X \rightarrow Y$ is continuous.  Prove that if $X$ is
compact, then $f(X)$ is compact.  Prove that if $X$ is connected, then
$f(X)$ is connected.  (Only assume the definition of compactness and
the characterization of continuity in terms of preimages of open sets
being open.  Can you do it with just the $\varepsilon$-$\delta$
definition of continuity?)
\item
Suppose that $f: X\rightarrow \R$ and $g: X \rightarrow \R$ are
continuous.  Define $h: X \rightarrow \R$ by $h(x) = \min(f(x),
g(x))$.  Must $h$ be continuous?
\item
Is there a continuous function $f: (0,1) \cup (1,2) \rightarrow \R$ with
$f(1-)=0$ and $f(1+)=3$?  Is there a uniformly continuous function
like this?
\end{enumerate}

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