\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