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\topmatter
\title Math 335, Test 1, Fall (1993) \endtitle
\endtopmatter

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.  The problems are worth
20~points apiece; in the case of a problem with multiple sections, the
credit will be divided equally between the parts unless otherwise
specified.  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{\bf#1.}\ }
\define\ritem#1{\item"(#1)"}

\prob1 (a) State the principle of mathematical induction.   (b) {\sl Using this
principle}, give a careful proof of the formula
$$1^2+2^2+\cdots+n^2=n(n+1)(2n+1)/6$$

\prob2 (a) State the Completeness Axiom for the field $\Bbb R$ of real
numbers.  (b) Explain what is meant by the ``Denseness of $\Bbb Q$''.


\prob3 The {\sl Archimedean Property} states that if $a>0$ and $b>0$
are real numbers, then for some positive integer~$n$ we have $na>b$.
Use the Completeness Axiom to prove this property.

\prob4  Prove using the definition of limit that
$\lim_{n\to\infty}a^n=0$ if $\vert a\vert<1$.


\prob5  Consider the set $\{s\in\Bbb Q\vert s^2<3\}$.
\roster \ritem a (5 pts)  Does this set have an upper bound in $\Bbb
Q$? If so, give one.
\ritem b (5 pts) Does this set have a least upper bound in $\Bbb Q$? If
so, determine what it is.
\ritem c (10 pts) Explain how your answers are consistent with the
Completeness Axiom.


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