\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 30 May, 2014  \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, 
$$
\lim_{n\to\infty} \int_0^1 \sqrt{\frac{nx}{1+n^2x^2}}\; dx.
$$

\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
$$
\lim_{n\ra\infty} \int_{-\pi}^\pi f(x)\cos(nx)\; dx = 0.
$$
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