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\newtheorem{theorem}{Theorem}
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\title{Math 335, Test 1, 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 (texbooks, 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)
Suppose that a set $A$ of real numbers is both open and closed. Prove
that either $A$ is the empty set or $A=\R$.
 

\prob2 (20 pts) 
Prove that the set $\R$ of real numbers is uncountable.




\prob3 (20 pts)
Suppose that $r$ is a real number. Prove (presumably by induction)
that the following formula is true for any integer $n\ge0$:
$$ 1+r+\cdots+r^n= \frac{1-r^{n+1}}{1-r}$$



\prob4 (a) (5~pts)
Define what it means for a subset $U$ of $\R$ to be a {\sl
neighborhood} of a real number~$x$. (b) (5~pts) Define what it means
for a subset $U$ of $\R$ to be {\sl open}. (c) Show that the set $\Q$
of rational numbers is not an open subset of $\R$.

\prob5 (a) (11~pts) State the Axiom of Completeness. (b) (9~pts) For
each of the following three conditions, either find a subset $A$ of
$\Bbb R$ which satisfies the condition, or explain why there is no
such subset.
\roster
\ritem i $A$ is nonempty and $\sup A$ does not exist.
\ritem {ii} $\sup A$ exists and $\sup A\notin A$.
\ritem {iii} $\sup A$ exists and $\sup A\in A$.
\endroster

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