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\centerline{Mathematics 242 -- Principles of Analysis}
\centerline{Makeup Exam 2 -- April 4, 2014}
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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)  Let $\{x_n\}$ be the sequence defined by $x_1 = 5$ and 
$$x_{n+1} = {12 + x_n^2\over 15}$$
for all $n \ge 1$.  Does this sequence converge?  Why?
If it does, what is the limit?
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II.  
\item{A)}  (15)  Show using the $\varepsilon, n_0$ definition that 
$$\lim_{n\to\infty} {4n + 1\over 7n + 3} = {4\over 7}.$$   
\item{B)}  (15)  Show using the $\varepsilon,\delta$ definition that
$$f(x) = {4x + 1 \over 7x + 3}$$
is continuous at $x = 0$.    
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III. (10) Suppose $\{x_n\}$ is a sequence such that 
$|x_n - 10| < 30$ for all $n \ge 1$.  Show that there exists
some number $a \in [-20,40]$ and a subsequence 
$\{x_{n_k}\}$ such that $x_{n_k} \to a$.  State any ``big theorems''
you are using.
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IV.  Give an example, or give a reason why there can be no
such examples:
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\item{A)} (10)  A function $f$ and an $a$ in the domain of $f$
such that $\lim_{x\to a^-} f(x)$ does not exist, but $\lim_{x \to a^+} f(x)$
does exist.   
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\item{B)} (10)  A sequence ${x_n}$ such that $x_n \to -4$, but 
for all $n_0 \in {\bf N}$ there exist $n \ge n_0$ with $x_n > 0$.  
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\item{C)} (10) A bounded function that is not continuous at any 
$c$ in its domain.  
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{\it Extra Credit.}  (10) 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 terms $x_n, x_{n+1}, x_{n+2}, x_{n+3}, x_{n+4}$ in the original sequence.  So for instance, 
$$y_1 = {x_1 + x_2 + x_3 + x_4 + x_5\over 5}, y_2 = {x_2 + x_3 + x_4 + x_5 + x_6\over 5}, $$
and so on.  Show that $\{y_n\}$ also converges to $a$.  
\bye
