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{\bf Review Sheet for Final}
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\noindent{\bf Format:} the format of the final will be similar to that
of the first and second midterms.  At least half the problems will concern 
material beyond that tested on the two midterms.  This review sheet
is directed at this portion of the exam.  At least one third of the final
will be devoted to older material.  The review sheets for the
midterms (as well as the midterms themselves!) should be useful for
refreshing your knowledge of that material.  All told, I'd like to make
the final equal in length to about 1.5 midterms.

\vspace*{.2in}
\noindent{\bf Things to know: }
\begin{description}
\item[Definitions and statements.] Upper bound, supremum (least upper
bound), lower bound, infimum, sequence, completeness axiom, Archimedean
Property, convergent sequence, bounded sequence, monotone sequence, 
Cauchy sequence, subsequence, Bolzano-Weierstrass Theorem, $k$-ary expansion, 
Squeeze Theorem.

\item[Proofs you should know:] (from new material) Bolzano-Weierstrass
Theorem, real numbers are uncountable.

\item[Computational skills.] 
General facility with inequalities; determining the supremum of a set (in
particular, whether it exists); checking whether a sequence is 
monotone; computing the $k$-ary expansion of a given rational number, and
conversely, determining a rational number from a given $k$-ary
expansion.

\item[Good topics for proofs.] 
Proofs concerning supremums/infimums.  Proofs using the definition of
a limit (i.e. of convergence of a sequence).  Proofs using the monotone
convergence theorem.

\item[Some problems about $k$-ary expansions to try.]  13.6, 13.14, 13.15, 
13.16, 13.18, 13.33, 13.37, 14.6, 14.7, 14.35, 14.36, 14.39.
\end{description}

\vspace*{.2in}
\noindent{\bf Advice for studying: }
The advice I gave concerning the first two exams still applies.  I don't
think I have much new to add this time.

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