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\centerline{Mathematics 242 -- Principles of Analysis}
\centerline{Information on Exam 2}
\centerline{October 20, 2004}
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\noindent
{\it General Information}
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The second hour exam for the course will be given in class on
Friday, October 29.  This will 
be an individual, in-class, closed book exam.  I will be happy 
to hold a late afternoon or evening review session to help you 
prepare.  Late afternoon times are possible Wednesday or Thursday,
but I will have to leave campus no later than 5:30pm to get to 
a rehearsal, so evenings will not work for me those days.  
Tuesday evening would be OK.
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\noindent
{\it Topics to be Covered}
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The exam will cover the material we have covered since the last 
exam, through the material on subsequential limits and the 
Bolzano-Weierstrass Theorem from class on Wednesday, October 20.
This is sections 11, 12, 16, 17, 18, 19 in the text, but not
all the topics in those sections were discussed in class. You
are responsible for only what we did talk about:
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\item{1)}  The algebraic and order properties of ${\bf R}$,
\item{2)}  The least upper bound axiom and consequences
(the Archimedean property, the ``density'' of the rational 
numbers -- that is the statement that for all real $x < y$,
there exists a rational number $r$ with $x < r < y$, and so forth)
\item{3)}  Sequences, convergence ($\lim_{n\to\infty} s_n = L$ -- both
the $\epsilon$ - $N$ definition, and computing limits via the 
Limit Theorems (See 17.1 and 17.4 in text)
\item{4)}  The Nested Interval Theorem, 
\item{5)}  Subsequences and the Bolzano-Weierstrass Theorem
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\noindent
{\it What to Expect}
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The exam will have four or five questions, each possibly with several
parts.  Some questions will ask for a precise statement of a 
definition or a theorem we have discussed.  Be prepared to give
careful statements of 
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\item{1)}  The Least Upper Bound Axiom
\item{2)}  The definition of convergence for a sequence
\item{3)}  The Bolzano-Weierstrass Theorem
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\noindent
Also know and be able to give these proofs:
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\item{1)}  Prove that a given sequence $x_n$ converges to a limit $L$, 
using the $\epsilon$ - $N$ definition
\item{2)}  The limit sum rule for sequences (Part 1 of 17.1 -- 
If $s_n \to L$ and 
$t_n \to M$ then $s_n + t_n \to L + M$)
\item{3)}  Every monotone increasing sequence of real numbers that is bounded
above converges.
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The other questions will be similar to questions from the problem sets
and discussions.  Some good review problems to look at are
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\item{} Section 11/3, 6, 8, 9; 
\item{} Section 12/3, 4, 6, 9; 
\item{} Section 16/4, 5, 6, 10, 11; 
\item{} Section 17/3, 5, 6 
\item{} Section 18/2, 3, 5
\item{} Section 19/3, 4, 5
\bye
