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\topmatter
\title Math 335, Final Exam, Fall  1995 \endtitle
\endtopmatter

The test will be 2 hours 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}
\define\Limit{\Cal L}
\define\part{{\parskip=0pt\par\noindent\hbox to 3ex{\hfil}\ }}

\prob1 (a) (20 pts) Suppose that $f$ is a continuous function from a
closed and bounded set $H$ into~$\R$. Prove that the range of~$f$ is also
closed and bounded.
\part(b) (10~pts) Let~$f$ be the function from the open interval $(0,1)$
into~$\R$ given by $f(x)=1/x$. Show that there does not exist a
continuous function on the closed interval $[0,1]$ which agrees
with~$f$ on  $(0,1)$.

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


\prob3 (20 pts) Prove that every convergent sequence is a Cauchy
sequence.

\prob4 Let $(x_n)$ be a sequence of real numbers. \part(a) (10~pts) Define
$\limsup x_n$.  \part(b) (5~pts) Give an example of a bounded sequence
$(x_n)$ with the property
that $\limsup x_n$ is not equal to $\liminf x_n$. \part(c) (5~pts) Give an
example of a bounded sequence $(x_n)$ with the property that there
exists a number in the sequence which is greater than $\limsup x_n$.

\prob5 (20~pts) Use induction to prove that for any integer $n\ge1$,
$$1^2+2^2+3^2+\cdots+n^2=n(n+1)(2n+1)/6\,\,.$$

\prob6 (a) (5 pts) Define the {\sl interior} of a subset $S$
of~$\R$. \part(b) (5~pts) Define what it means for a subset $S$ of $\R$
to be {\sl open}; define what it means for~$S$ to be {\sl closed}. \part(c) (5~pts) Find a subset of $\R$ which is
neither open nor closed. \part(d) (5~pts) Find a subset of $\R$ which is
not open, is not closed, and has empty interior. 

\prob7 (20~pts) Suppose that $a>0$ is a real number and that $(x_n)$
is a sequence of points in~$\R$ with $\lim
x_n=a$. Prove that there exists a number~$N$ such that if $n>N$ then
$x_n>0$.




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