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\begin{document}

\title{Analysis II}
\author{Problem Set 6}
\date{Midterm Review Problems -- Not for Credit}
\maketitle

\begin{enumerate}

\item \emph{Topic:} Integration by parts, asymptotics, Laplace's
method (idea only)

Show that, as $x \to \infty$,
\[
  \int_x^\infty \e^{-x^4} \, \d x
  \sim \frac{\e^{-x^4}}{4 \, x^3} \,.
\]

\item \emph{Topic}: Elementary point-set topology in metric spaces.

\begin{enumerate}
\item Let $A$ be an open subset and $B$ be a closed subset of a metric
space.  Determine whether the following occur always, sometimes, or
never; give proofs or (counter)-examples.

\begin{itemize}
\item $\overline{A \cup B}$ is open.
\item $\overline{A \cup B}$ is closed.
\item $\overline{A \cup B}$ is both open and closed.
\item $\overline{A \cup B}$ is neither open nor closed.
\end{itemize}

\item
Let $X$ be a metric and $f$ be a continuous real-valued function on
$X$.  Prove that the zero-set of $f$, i.e., the set of all points $p
\in X$ such that $f(p)=0$, is closed.
\end{enumerate}

\item \emph{Topic}: Compactness

\begin{enumerate}
\item
Let $\{E_\alpha\}$ be a family of compact subsets of a metric
space $X$.  Show that
\[
  E = \bigcap_\alpha E_\alpha
\]
is compact.

\item Prove that every compact metric space has a countable dense
subset.
\end{enumerate}

\item \label{1} \emph{Topic}: Uniform convergence (elementary
properties and examples)

Let $f_n \colon [0,1] \to \R$ and
$g_n\colon [0,1] \to [0,1]$ be sequences of continuous functions
con\-ver\-ging uniformly to $f \colon [0,1] \to \R$ and
$g \colon [0,1] \to [0,1]$, respectively.  Show that
\[
  f_n \circ g_n \to f \circ g
\]
uniformly as $n \to \infty$.

\item \emph{Topic}: Arzel\`a--Ascoli theorem

Construct a proof of \emph{Peano's existence theorem} in the
following simplified version.  Suppose $f \in C([-1,1])$.  Then there
exists $\delta>0$ such that the differential equation
\begin{gather*}
  y'(t) = f(y(t)) \,, \\
  y(0) = 0
\end{gather*}
has a solution, not necessarily unique, on the interval $[0,\delta]$.

Proceed as follows.  For every $n \in \mathbb N$, define the function
$y_n(t)$ via the recursive process
\begin{gather*}
  y_n(t) =
  \begin{dcases}
    0 & \text{for } t \in [0,\delta/n] \,, \\
    \int_0^t f(y_n(s-\delta/n)) \, \d s & 
    \text{for } t \in (\delta/n,\delta] \,.
  \end{dcases}
\end{gather*}

\begin{enumerate}
\item Find $\delta>0$ such that $\{y_n\}$ is well-defined.
\item Show that $\{y_n\}$ is point-wise bounded, cf.\ step (a), and
equicontinuous.  

\item Argue that there exists a subsequence converging uniformly to
some $y \in C([0,\delta])$.

\item Show that $y$ satisfies the integral equation
\[
  y(t) = \int_0^t f(y(s)) \, \d s \,.
\]
(You may appeal to the result of Problem~\ref{1} above.)

\item Argue that this implies that $y$ satisfies the differential
equation in Peano's existence theorem.

\end{enumerate}


\item \emph{Topic}: Weierstra\ss\ polynomial approximation theorem

Show that if $f$ is continuous on $\R$, there exists a sequence
of polynomials $\{p_n\}$ that converges to $f$ uniformly on every
bounded subset of $\R$.


\item \emph{Topic}: Power series expansions and radius of convergence

Find a power series expansion centered at $0$ for
\[
  \frac1{(1+x^2)^2}
\]
and determine the radius of convergence.

\end{enumerate}

\end{document}
