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\centerline{Mathematics 242 -- Principles of Analysis}
\centerline{Information on Exam 1}
\centerline{September 29, 2000}
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\noindent
{\bf General Information}
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The first exam for the course will be given in class on
Friday, October 6.  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.  Wednesday, October 4 will probably be the best day for 
that.
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\noindent
{\bf Topics to be Covered}
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The exam will cover the material we have covered since the start of the 
semester up to and including class on September 27:
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\item{1)}  The real number system, rational and irrational numbers, 
the algebraic and order properties, the least upper bound axiom.
\item{2)}  Mathematical induction
\item{3)}  Sequences, convergence ($\lim_{n\to\infty} x_n$ -- both
$\epsilon$ - $n_0$ definition, and computing limits via the 
``Big Limit Theorem''.
\item{4)}  Subsequences, The Nested Interval Theorem, and the Bolzano-Weierstrass Theorem
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\noindent
{\bf 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 proofs such as:
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\item{1)}  Prove that a given sequence $x_n$ converges to a limit $a$, 
using the $\epsilon$ - $n_0$ definition
\item{2)}  The limit sum rule for sequences (Part a of the 
``Big Limit Theorem'' -- If $x_n \to a$ and 
$y_n \to b$ then $x_n + y_n \to a + b$)
\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 1.1/6,8,9; 
\item{} Section 1.3/8,9; 
\item{} Section 1.4/4,13; 
\item{} Section 1.5/1,2,7,8;
\item{} Section 2.1/1,2,3,12;
\item{} Section 2.2/1,3,17,19; 
\item{} Section 2.3/1,2,9; 
\item{} Section 2.5/1,2,5,12
\bye
