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\lhead{\scriptsize MA4510: Introduction to Analysis (Fall 2010)} 
\chead{} 
\rhead{\scriptsize Exam 1} 
\lfoot{\scriptsize This work is licensed under the \href{http://creativecommons.org/licenses/by-sa/3.0/us/}{Creative Commons Attribution-Share Alike 3.0 License}.} 
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\rfoot{\scriptsize Written by \href{http://dcernst.wikidot.com}{D.C. Ernst}} 
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\begin{center}

{\Large\bf MA 4510: Introduction to Analysis (Fall 2010)}\\
\smallskip
{\Large\bf Exam 1}

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  \fbox{\parbox{7in}{
    \vspace{12pt}
    \textbf{\large NAME:}
    \vspace{12pt}
  }}

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\section*{Instructions}

Complete all of the problems in Part 1.  For Part 2, prove any 4 theorems.

\bigskip

This exam is worth 100 points and 15\% of your overall grade.  %Your written presentation of the proofs (which includes spelling, grammar, punctuation, clarity, and legibility) is worth 10 points.

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I expect your proofs to be \emph{well-written, neat, and organized}.  You should write in \emph{complete sentences}.  Do not turn in rough drafts.  What you turn in should be the ``polished'' version of potentially several drafts.  Feel free to type up your final version.  

\bigskip

The \LaTeX\ source file of this exam is also available if you are interested in typing up your solutions using \LaTeX.  I'll be happy to help you do this.

\bigskip

The simple rules for this portion of the exam are:

\begin{enumerate}
\item You may freely use any theorems that we have discussed in class, but you should make it clear where you are using a previous result and which result you are using.  For example, if a sentence in your proof follows from Theorem 1.41, then you should say so.
\item You cannot use any results from the book or otherwise that we have not covered, unless you prove them.
\item You are NOT allowed to copy someone else's work.
\item You are NOT allowed to let someone else copy your work.
\item You are allowed to discuss the problems with each other and critique each other's work.
\end{enumerate}

The exam is due by 11:15\textsc{am} on Wednesday, October 13.  You should turn in this cover page and all of the work that you have decided to submit.

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To convince me that you have read and understand the instructions, sign in the box below.

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  \fbox{\parbox{7in}{
    \vspace{12pt}
    \textbf{\large Signature:} \hfill
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  }}

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Good luck and have fun!

\newpage

\section*{Part 1}
Provide an example of each of the following.  You do \emph{not} need to justify your answers.  (5 points each)

\begin{enumerate}

\item A sequence of distinct points that converges to 42.

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\item A point set that has a leftmost point and a first point to the right.

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\item A point set that has exactly two limit points.

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\item A sequence that is bounded but does not converge.

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\item A sequence that is increasing but does converge.

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\item An open point set that is not an open interval.

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\item A closed point set that is not bounded above or below.

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\item A point set that is neither open or closed.

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\item A sequence $p_1, p_2, p_3,\ldots$ such that $|p_1|, |p_2|, |p_3|,\ldots$ converges $x$ but $p_1, p_2, p_3, \ldots$ does not converge to $x$.

\vfill

\item A point set that is not bounded above or below and does not contain infinitely many of its limit points.

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\item A point set $M$ that has a limit point $p$ such that there exists a closed interval $[a,b]$ containing $p$ that does not contain any other points in $M$.\footnote{This is essentially Problem 1.62.}

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\item A point set with a left-most point and right-most point that is not closed and bounded.

\vfill

\end{enumerate}

\newpage

\section*{Part 2}
Prove any 4 of the following theorems.  (10 points each)

\begin{theorem}
If the sequence $p_1,p_2,p_3,\dots$ converges to the point $x$, and, for each positive integer $n$, $p_n \ne p_{n+1}$, then $x$ is a limit point of the set that is the range of the sequence.\footnote{This is Problem 1.33.}
\end{theorem}

\newpage

\begin{theorem}
If the sequence $p_1,p_2,p_3,\dots$ converges to $x$ and the sequence $q_1,q_2,q_3,\dots$ converges to $y$, then the sequence $p_1+q_1,p_2+q_2,p_3+q_3,\dots$ converges to $x+y$.\footnote{This is Problem 1.36.}
\end{theorem}

\newpage

\begin{theorem}
Consider a sequence $p_1, p_2, p_3, \ldots$.  If the sequence $|p_1|, |p_2|, |p_3|,\ldots$ converges to 0, then $p_1, p_2, p_3, \ldots$ converges to 0.
\end{theorem}

\newpage

\begin{theorem}
If $M$ has $p$ as a limit point, then there exists either an increasing or a decreasing sequence of points of $M$ converging to $p.$\footnote{This is Theorem 1.64.}
\end{theorem}

\newpage

\begin{theorem}
If the sequence $p_1, p_2, p_3, \ldots$ is increasing and bounded, then the sequence converges.
\end{theorem}

\end{document}
