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\centerline{Mathematics 242 -- Principles of Analysis}
\centerline{Exam 1}
\centerline{February 22, 2013}
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{\it Directions:}  Do all work in the blue exam booklet (do not place any work that you want considered for credit on this sheet).  There are 100 regular and 10 extra credit points.
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I.  
\item{A)} (10)  Let 
$$A = \bigcup_{n=1}^\infty \left({1\over n} ,2 - {1\over n}\right)$$
Explain why $A$ is bounded and determine upper and lower bounds for $A$. 
\item{B)} (10)  {\it Define:}  The real number $a$ is a {\it least upper bound} of $A \subset {\bf R}$, and
state the Least Upper Bound Axiom for ${\bf R}$.
\item{C)} (10)  Let $A$ be a bounded subset of ${\bf R}$ and let $B = \{4\cdot x - 3\mid x \in A\}$.
What can be said about ${\rm lub}(B)$?  Prove your assertion.
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II.  (20) Let $x_n$ be the sequence defined by the rules $x_1 = 1$ and $x_{n+1} = {1\over 3} x_n + 1$
for all $n \ge 1$.  Show by mathematical induction that 
$$x_n = {1 - {1\over 3^n} \over 1 - {1\over 3}}\ {\rm for\ all\ } n\ge 1.$$
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III.   Let $x_n = {5^n \over 7^n + 3}$ for all natural numbers $n \ge 1$.  
\item{A)}  (10) Determine $\lim_{n\to\infty} x_n$ intuitively.
\item{B)}  (20)  Use the $\varepsilon$, $n_0$ definition of convergence
to prove that $\{x_n\}$ converges to the number you identified in part A.  
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IV.  True-False.  For each true statement, give a short proof or reason.  For each false
statement give an explicit counterexample.
\item{A)} (10)  Let $A$ be a bounded set of real numbers and let $a = {\rm lub}(A)$.
Then for each $\varepsilon > 0$, there exists $x\in A$ such that $a - \varepsilon < x < a$.  
\item{B)} (10)  If $r$ is a nonzero rational number and $s$ is a nonzero irrational
number, then $r/s$ is irrational.
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{\it Extra Credit} (10)  Is it possible to produce a sequence $x_n$ whose
terms include all the rational numbers $p/q$ with $p, q\in {\bf Z}$ and 
$p,q > 0$?   If so, give an indication how to construct such a sequence.
If not, give a reason why there cannot exist such a sequence.
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