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\title{Analysis II}
\author{Midterm Exam}
\date{March 18, 2019}
\maketitle

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\begin{enumerate}

\item
Show that, as $x \to \infty$,
\[
  \int_x^\infty \frac{\e^{-x}}x \, \d x
  \sim \frac{\e^{-x}}x \,.
\]
\hfill(5)


\item 
\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[(i)] $A \setminus B \equiv \{ x \in A \colon x \notin B\}$ is open.
\item[(ii)] $A \setminus B$ is closed.
\item[(iii)] $A \setminus B$ is both open and closed.
\item[(iv)] $A \setminus B$ is neither open nor closed.
\end{itemize}

\item Let $X$, $Y$, and $Z$ be metric spaces, and $f \colon Y \to Z$
and $g \colon X \to Y$ be continuous mappings.  Use the topological
characterization of continuity to show that the composition
$f \circ g$ is continuous.
\end{enumerate}
\hfill (5+5)


\item 
\begin{enumerate}
\item
Let $\{x_n\}$ be a converging sequence of elements in a
metric space $X$ with limit $x$.  Show that the set
\[
  E = \{x\} \cup \{ x_n \colon n \in \mathbb N \}
\]
is compact.


\item Prove that the boundary of a compact set is compact.
\end{enumerate}
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\item Let $f_n, g_n \colon \R \to \R$ be sequences of bounded
continuous functions which converge uniformly to functions $f$ and
$g$, respectively.
\begin{enumerate}
\item Show that $f_n \, g_n \to f \, g$ uniformly as $n \to \infty$.
\item Give an example which shows that convergence may fail to be
uniform when the assumption of boundedness is dropped.
\end{enumerate}
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\item Consider the sequence of functions on $[0,1]$ defined by
\[
  f_n(x) = n \, \sin \frac{x}n \,.
\]
\begin{enumerate}
\item Show that there exists a uniformly converging subsequence of
$\{f_n\}$.

\item What is the limit function?
\end{enumerate}
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\item Let $f \in C([0,1])$.  Show that there exists a sequence of
polynomials $\{p_n\}$ satisfying $p_n(0) = f(0)$ such that $p_n \to f$
uniformly on $[0,1]$.
\hfill(5)


\item Find a power series expansion centered at $0$ for
\[
  \ln(1+x)
\]
and determine the radius of convergence.
\hfill(5)


\end{enumerate}

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