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\topmatter
\title Math 335, Final Exam, Fall (1993) \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.
Except for problem~5, each problem is worth 20~points.
Unless otherwise indicated, credit on a multipart problem
is divided evenly between the parts.
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.}}\ }
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\define\ritem#1{\item"(#1)"}
\define\dom{\operatorname{dom}}
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\prob1 (a) State the principle of mathematical induction.
(b) 
Show that for every integer $n\ge1$ there is an equality
$$1+2+3+\cdots+n =\frac{n(n+1)}2\,\,.$$

\prob2 (a) Explain what is meant by the ``denseness of ${\Bbb
  Q}$''. (b) Suppose that $f$ is a continuous function on
${\Bbb R}$ such that $f(x)=0$ for all $x\in\Bbb Q$. Show that $f$ is
the zero function.

\prob3 Prove that every bounded monotone sequence 
converges.

\prob4 (a) State the comparison test for the convergence of
series. (b)  Show that if $\sum a_n$ is a convergent series of
nonnegative numbers and $p>1$, then $\sum a_n^p$ converges.

\prob5 (a) (20 pts) Prove that every sequence $(s_n)$ has a monotone
subsequence. (b) (10 pts) State and prove the Bolzano-Weierstrass
theorem.

\prob6 Suppose that $f$ is uniformly continuous on a set~$S$
and that $(s_n)$ is a Cauchy sequence in~$S$. Prove that $(f(s_n))$ is
a Cauchy sequence.

\prob7 
\roster
\ritem a Suppose that $(s_n)$ is a sequence and $s$ is a real
number. Give the definition of what it means for the limit of $(s_n)$
to be equal to~$s$.
\ritem b Suppose that $f$ is a function defined in some
interval containing the real number $a$. Define in terms of sequences
what it means for $f$ to be continuous at $a$.
\ritem c Suppose that $f$ is a function defined in some
interval containing the real number $a$. Define in terms of epsilons
and deltas what it means for $f$ to be continuous at $a$.
\ritem c Let $f$ be a function defined for all real numbers,
and let $b$ be a real number. Define what ``$\lim_{x\to\infty}f(x)=b$''
means.
\endroster
\enddocument



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