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\centerline{\bf Ph.D. Qualifying Exam}
\bigskip
\centerline{Fall 2004}
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\noindent {\bf Instructions:}
\begin{itemize}
\item[1.] If you think that a problem is incorrectly stated ask the proctor. If his explanation is not to
your satisfaction, interpret the problem as you see fit, but not so that the answer is trivial.
\item[2.] From each part solve 3 of the 5 five problems.
\item[3.] If you solve more than three problems from each part, indicate the problems that you wish to have graded.
\end{itemize}
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\centerline{\bf Part A}
\medskip\noindent
1. Suppose that $a_n>0$, show that $\ds \sum_{n=1}^\infty a_n$ converges if and only if $\ds
\sum_{n=1}^\infty\frac{a_n}{1+a_n}$ converges.
\medskip\noindent
2. Let $A$ be a closed and bounded subset of $C_2([0,1])$, the twice continuously differentiable functions with the
supremum norm . Show that A is precompact subset of $C_1([0,1])$.
\medskip\noindent
3. Consider the closed unit ball B in $C_0([0,1])$, show that B cannot be covered by a finite number of balls of
radius $r$ where $r<1$.
\medskip\noindent
4. Suppose that $f_n(x)$ is a decreasing sequence of upper semi-continuous function with pointwise limit $f(x)$.
Show that $f(x)$ is upper semi-continuous.
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5. If $\sum a_n$ is a divergent series of positive terms and $s_n$ denotes its $n^\mathrm{th}$ partial sum, show
that the series $\sum\ds\frac{a_n}{s_n^2}$ converges.
\bigskip
\centerline{\bf Part B}
\medskip\noindent
1. Consider the sequence of functions $\ds f_n(x)=\frac{ne^{x^2}}{1+n^2x^2}$. Compute $\lim_{n\to\infty}\int_0^1
f_n(x)dx$. Justify each of your steps carefully.
\medskip\noindent
2. Suppose that $E \subset \mathbf{R}$ is Lebesgue measurable and that there is a real number $\alpha$, $0\leq
\alpha < 1$ such that for any interval $I$, $\mu(E\cap I) \leq \alpha\mu(I)$ where $\mu$ is Lebesgue measure. Show
that $\mu(E)= 0$.
\medskip\noindent
3. Suppose that $\{f_n(x)\}$ is a sequence of real valued measurable functions of a real variable. Suppose that
there is an integrable function $g$ with $|f_n(x)|