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\centerline{Mathematics 242 -- Principles of Analysis}
\centerline{Exam 1 -- October 5, 2005}
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{\it Directions: } Do all work in the blue exam book.  
There are 100 possible regular points and 10 possible
Extra Credit points.
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I.  
\item{A)}  (10) State the Axiom of Completeness for the real number system.
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\item{B)}  (10) Show that if $s = \sup(A)$ for some $A\subset {\bf R}$
and $\varepsilon > 0$, then there exists some $a \in A$ satisfying
$s - \varepsilon < a \le s$.  
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\item{C)}  (10) Let $A = \cup_{n=1}^\infty \left(0,1-{1\over 2n}\right)$.
Explain why $A$ is bounded above and determine $\sup(A)$.  
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II. 
\item{A)} (5) Find 
$$\lim_{n\to\infty} {2n^2 - 1\over 5n^2 + 2n}$$  
using the Algebraic Limit Theorem.
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\item{B)} (15) Prove that $x_n = {2n^2 - 1\over 5n^2 + 2n}$ 
converges to your limit from part A {\it using the definition}.
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III.  Let $(x_n)$ be the sequence defined by $x_1  = 2$ and
$x_{n+1} = {2x_n - 1\over 3}$ for all $n \ge 1$.  
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\item{A)} (15) Using mathematical induction, show that 
$x_n \ge -1$ for all $n\ge 1$.  
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\item{B)} (15) Using mathematical induction, show that 
$(x_n)$ is strictly decreasing:  $x_{n+1} < x_n$ for all $n\ge 1$.
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IV.  True-False.  For each true statement give a short proof or 
reason; for each false statement give a counterexample.
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\item{A)}  (10) If $A$ and $B$ are sets of real numbers 
with the property that $a > b$ for all $a\in A$ and $b \in B$,
then $\inf(A) > \sup(B)$.
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\item{B)}  (10) If $r$ is a nonzero rational number and $t$
is an irrational number, then $rt$ is irrational.
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{\it Extra Credit}  (10)  {\it Caution: this problem may be ``habit forming.''
Only attempt after finishing the rest of the exam!} 
Can you find a 1-1, onto function $f : {\bf R} \to [0,1]$?  If
so, describe one by giving a formula or a graph.  If not, say why not.


\bye