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\newtheorem{theorem}{Theorem}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{definition}{Definition}

\title{Math 335, Test 2, Fall  1995}

\begin{document}


The test will be 50 minutes in length.  Please write your name on the
cover of your blue book and write the solutions inside; start the
solution to each problem on a new page.  Each problem has the point
value indicated.  This test is being administered under the provisions
of the Honor Code. Your work should be your own, and you should not
make use of any outside material (textbooks, notes) during the test.
What you write should be neat, grammatical, clear and concise. Good
luck.


%\define\prob#1{\par\medskip\noindent{\bf#1.}\ }
\define\prob#1{\par\medskip\noindent\hbox to 3ex{\bf#1.\hfil}}
%\define\fakeprob#1{\par\noindent\phantom{{\bf #1.}}\ }
\define\fakeprob#1{\par\noindent\hbox to 3ex{\hfil}}
\define\ritem#1{\item"(#1)"}
\define\dom{\operatorname{dom}}
\def\ourstrut{\vrule width 0.0pt depth 0.0pt height 15.0pt\relax}
\define\R{\Bbb R}
\define\Q{\Bbb Q}

\prob1 (20 pts)
Prove that any bounded sequence of real numbers has a partial limit
(i.e., a convergent subsequence).
 

\prob2 (20 pts) 
Prove that any bounded increasing sequence of real numbers converges.

\prob3 
(a) (5 pts) Define what it means for a subset $A$ of~$\R$ to be {\sl
closed}.
(b) (15~pts) Suppose that $A$ is a closed subset of~$\R$ and that
$(x_n)$ is a convergent sequence such that $x_n\in A$ for
all~$n$. Show that $\lim x_n\in A$.


\prob4 
(a) (5 pts) Give the ``$\epsilon\text{--}N$'' formulation of what it
means for a sequence $(x_n)$ to converge to a limit~$x$.
(b) (5~pts) Using (a), write down in concrete terms what it means for
a sequence $(x_n)$ {\bf NOT} to converge to~$x$.
(c) (10~pts). Let $x_n=1/n^2$. Show carefully using~(a) that $(x_n)$
converges to~$0$.


\prob5 
(a) (5 pts) Define what it means for a sequence $(x_n)$ to be a Cauchy
sequence.
(b) (15~pts) Determine whether or not the following sequences $(x_n)$
are Cauchy sequences:
\roster
\ritem i $\displaystyle x_n= (-1)^n + \frac1n$
\ritem {ii} \ourstrut$\displaystyle x_n=1+ \frac{(1)^n}n$
\ritem {iii} \ourstrut$\displaystyle x_n=\log(n)$
\endroster

\end{document}



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