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\centerline{Mathematics 242 -- Principles of Analysis}
\centerline{Makeup 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 $n$ is any natural number, then there exists some $a \in A$ satisfying
$s - {1\over n} < a \le s$.  
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\item{C)}  (10) Let $A = \cap_{n=1}^\infty \left(1,2+{1\over n}\right)$.
Explain why $A$ is bounded above and determine $\sup(A)$.  
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II. 
\item{A)} (10) Find 
$$\lim_{n\to\infty} {n+1\over 4n^2 + 7n}$$  
using the Algebraic Limit Theorem.
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\item{B)} (15) Prove that $x_n = {n+1\over 4n^2 + 7n}$ 
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 $|x_n - (-1)| = 3\left({2\over 3}\right)^{n-1}$ 
for all $n \ge 1$.
\item{B)} (10)  Deduce from part A that $\lim_{n\to\infty} x_n = -1$.
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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 
that are bounded above and $AB$ is defined to be the set
of all products $ab$ for $a\in A$ and $b\in B$, then
$\sup(AB) = \sup(A)\sup(B)$.
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\item{B)}  (10) If $r$ is a nonzero rational number and $t$
is an irrational number, then $t/r$ 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