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\centerline{MATH 242 -- Principles of Analysis}
\centerline{Exam 2 -- April 1, 2011}
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{\it Directions}
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Do all work in the blue exam booklet.  There are 100 possible regular
points and 10 Extra Credit Points.
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I.  
\item{A)} (20) State and prove the Monotone Convergence Theorem
for sequences.  (You may give the proof in the case that the 
sequence is monotone increasing.)
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\item{B)} (10)  Suppose that $x_n$ is a sequence of strictly 
{\it negative} numbers and $x_{n+1}/x_n \le 1$ for all $n \ge 1$. 
Show that $\lim_{n\to\infty} x_n = a$ exists in ${\bf R}$
and satisfies $a \le 0$.
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II. (20) Suppose $\{x_n\}$ is a sequence such that 
$|x_n - 10| < 100$ for all $n \ge 1$.  Show that there exists
some number $a \in [-90,110]$ and a subsequence 
$\{x_{n_k}\}$ such that $x_{n_k} \to a$.  State any ``big theorems''
you are using.
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III.  (20) Show using the $\varepsilon,\delta$ definition that 
$\lim_{x\to 2} x^2 - x = 2$. 
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IV. True/False.  For each true statement, give a short proof.
For each false statement, give a counterexample or reason.  
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\item{A)} (10) If $\{x_n\}$ diverges, then for all $a\in {\bf R}$,
there exist $\varepsilon_0 > 0$ and $n_0 \in {\bf N}$ such that
$|x_n - a| \ge \varepsilon_0$ for all $n \ge n_0$.  
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\item{B)} (10) It is possible to find a function $f: {\bf R} \to {\bf R}$ that 
is continuous at $c = e$, with $f(e) = 0$, but
satisfying $f(x) = 1$ for all rational $x$.
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\item{C)}  (10)  The sequence defined by $x_1 = 1$
and $x_n = \sqrt{3x_{n-1} + 1}$ for $n\ge 2$ is monotone increasing. 
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{\it Extra Credit.}  (10) 
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Assume that $\{x_n\}$ is a sequence that 
converges to $a$.  Construct a new sequence $\{y_n\}$
by making $y_n$ the average of the first $n$ terms in the original sequence:
$y_n = {x_1 + x_2 + \cdots + x_n\over n}.$
Show that $\{y_n\}$ also converges to $a$.  
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