\documentclass[12pt]{article}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsmath}
\textheight= 9 in
\topmargin=-0.5 in
\textwidth=6in
\oddsidemargin= 0.25 in
\thispagestyle{empty}
\def\ra{\rightarrow}
\def\Reals{{\mathbb R}}
\def\Rationals{{\mathbb Q}}
\def\abs#1{\left|#1\right|}
\begin{document}
\thispagestyle{empty}
\Large \noindent \underline{\textbf{SAMPLE}} \hfill\underline{\textbf{SAMPLE}}
\scriptsize \noindent 31 May, 2010 \hfill \tiny [AUTHOR: Maxwell/Bueler]
\normalsize\bigskip
\centerline{\large\textbf{Real Analysis Comprehensive Exam}}
\bigskip
Complete {\bf EIGHT} of the following ten problems. It is better to fully
complete fewer problems than to earn partial credit on many problems.
\bigskip
\begin{enumerate}
\item Compute, with justification,
$$
\lim_{n\ra\infty} \int_0^1 \frac{nx}{1+n^2x^2}\; dx.
$$
Hint: $(1-nx)^2 \ge 0$.
\item Give examples of the following
\begin{enumerate}
\item A sequence $(f_n)$ in $C[0,1]$ that converges pointwise to $0$
but such that $\int_0^1 f_n\not\ra 0$.
\item A bounded sequence in $\ell_1$ that has no convergent subsequence.
\item A sequence in $C[0,1]$ that converges pointwise to a discontinuous function.
\end{enumerate}
\item Let $(f_n)$ be a bounded sequence of functions in $C[0,1]$.
Define
$$
F_n(x) = \int_0^x f_n(s)\; ds.
$$
Show that $(F_n)$ has a uniformly convergent subsequence.
\item
\begin{enumerate}
\item State the Axiom of Completeness for $\Reals$.
\item Prove that a monotone increasing sequence of real numbers converges if and only
if it is bounded above.
\end{enumerate}
\item Let $m^*$ denote Lebesgue outer measure. Suppose that $E$ is a subset of $\Reals$
such that $m^*(E\cap (a,b)) \le 3/4 (b-a)$ for every interval $(a,b)$. Prove that
$m^*(E)=0$.
\item Let $(a_n)$ be a bounded sequence. Show that $\sum_{n=1}^\infty a_n e^{-nx}$
defines a continuous function on $[1,2]$.
\item Suppose that $X$ is compact and $f:X\ra \Reals$ is continuous. Prove that $f$ is
uniformly continuous.
\item Suppose $f\in L^1(\Reals)$ is uniformly continuous. Show that $\lim_{x\ra\infty}f(x)=0$.
\item
Let $f\in L^1[-\pi,\pi]$. Show that
$$
\lim_{n\ra\infty} \int_{-\pi}^\pi f(x)\cos(kx)\; dx = 0.
$$
Hint: First consider the case where $f$ is the characteristic function of
an interval.
\item Let $f:[0,1]\ra \Reals$ be an increasing, but not necessarily
continuous function. Show that $f$ is Borel measurable.
\end{enumerate}
\end{document}
\item Let $f:X\ra Y$ be a function and $\mathcal A$ a $\sigma$-algebra of sets in $X$.
Show that $\mathcal{B} = \{ B\subseteq Y : f^{-1}(B)\in \mathcal A\}$ is a $\sigma$-algebra
of sets in $Y$.
\item Suppose that $\{f_n\}$ is a sequence of functions in $B([0,1])$ (i.e. the bounded real-valued functions on $[0,1]$) converging uniformly to $g$. Suppose for some $x\in [0,1]$ that each $f_n$
is continuous at $x$. Show that $g$ is continuous at $x$.
OR