\magnification=\magstep1
\nopagenumbers
\centerline{Mathematics 242 -- Principles of Analysis}
\centerline{Exam 1 -- February 21, 2014}
\bigskip
\noindent
{\it Directions:}  Do all work in the blue exam booklet (do not place any work 
that you want to have considered for credit on this sheet).  There are 100 regular and 
10 extra credit points.
\bigskip
\noindent
I.
\item{A)} (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 ${\bf R}$. 
\bigskip
\item{B)} (10) Let
$$A = \bigcap_{n=1}^\infty \left(1 + {1\over n} , 4 - {1\over n}\right)$$
Explain why $A$ is bounded and determine the least upper and greatest lower bounds for $A$.
\bigskip
\item{C)} (15)  Let $A$ be a bounded subset of ${\bf R}$ with ${\rm lub}(A) = 2$,
and let $B = \{-4 x + 3 \mid x \in A\}$.
What can be said about ${\rm glb}(B)$?  Prove your assertion.
\bigskip
\noindent
II.  (15) Let $x_n$ be the sequence defined by the rules $x_1 = 1$ and $x_{n+1} = -{2\over 5} x_n + 1$
for all $n \ge 1$.  Show by mathematical induction that
$$0 \le x_n \le 1\ {\rm for\ all\ } n\ge 1.$$
\bigskip
\noindent
III.   Let $x_n = {3 n^2 \over 5n^2 + 3n + 1}$ for all natural numbers $n \ge 1$.
\bigskip
\item{A)}  (10) Determine $\lim_{n\to\infty} x_n$ intuitively.
\bigskip
\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.
\bigskip
\noindent
IV.  True-False.  For each true statement, give a short proof or reason.  For each false
statement, give an explicit counterexample.
\bigskip
\item{A)} (10)  If $A$ and $B$ are two nonempty bounded sets of real numbers and 
${\rm lub}(B) > {\rm lub}(A)$, then $y > x$ for all $y \in B$ and all $x\in A$.
\bigskip
\item{B)} (10)  Let 
$$A = \{x \in {\bf R} \mid 0 < x < 1,\ {\rm and }\ x = r\sqrt{2}\ {\rm for\ some}\ 
r \in {\bf Q}\}.$$
Then ${\rm lub}(A) = 1$.  
\bigskip
\noindent
{\it Extra Credit} (10)  Is it possible to produce a sequence $x_n$ whose
terms include all the positive and negative integers? 
If so, give an indication how to construct such a sequence.
If not, give a reason why there cannot exist such a sequence.
\bye
