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\centerline{Mathematics 242 -- Principles of Analysis}
\centerline{Exam 1 -- February 25, 2011}
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{\it Directions: } Read all the questions {\it carefully} before
starting to work.  Do all work in the blue exam booklet.  
There are 100 possible regular points and 10 possible
Extra Credit points.
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I.  
\item{A)}  (10) State the LUB Axiom for the real number system.
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\item{B)}  (10) Prove that if $A$ and $B$ are bounded sets of real numbers
with ${\rm lub}(A) < {\rm lub}(B)$, then there exists a single $y \in B$ satisfying
$y > x$ for all $x \in A$.    
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\item{C)}  (10) Let $A = \bigcap_{n=1}^\infty \left({-1\over 2n},1 + {1\over 2n}\right)$.
Explain why $A$ is bounded below and determine ${\rm glb}(A)$.  
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II. 
\item{A)} (20)  Prove that every interval $(a,b)$ with $0 < a < b$ contains
a rational number ${m\over n}$.
\item{B)}  (5) Use part A to show that every interval $(a,b)$ with $a < b < 0$
contains a rational number ${m\over n}$.    
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III. (15)  Let $f : {\bf N} \to {\bf R}$ be defined by $f(1)  = 2$ and
$f(n+1) = {2f(n) - 1\over 3}$ for all $n \ge 1$.    Using mathematical 
induction, show that $f(n) > -1$ 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 all $b \in B$,
then ${\rm glb}(A) > {\rm lub}(B)$.
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\item{B)}  (10) For all $n \ge 1$, $\sum_{k=0}^n {n\choose k} {1 \over 2^k} = {3^n\over 2^n}$.  
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\item{C)}  (10) The smallest $c$ such that 
$\{x \in {\bf R} \mid |x + 1| + |x + 10| = c\}$ is not empty is $c = 11$.    
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{\it Extra Credit}  (10)  
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{\it Caution: this problem may be ``habit forming.''
Only attempt after finishing the rest of the exam!} 
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Can you find a 1-1, onto function $f : {\bf R} \to (-1,1]$?  If
so, describe one by giving formulas or a graph.  If there is 
no such function, prove it.


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