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\centerline{Mathematics 242 -- Principles of Analysis}
\centerline{Exam 2 -- April 5, 2013}
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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)}  (15)  Show using the $\varepsilon, n_0$ definition that 
$$\lim_{n\to\infty} {5n + 1\over 3n + 3} = {5\over 3}.$$   
\item{B)}  (15)  Show using the $\varepsilon,\delta$ definition that
$$\lim_{x\to 1} {5x + 1 \over 3x + 3} = 1.$$  
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II.  
\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 $x_1 = 1$ and 
$$x_{n+1} = {5 x_n + 3\over 16}$$
for all $n \ge 1$.  Does this sequence converge?  Why?
If it does, what is the limit?
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III. (10)  Let $x_n = \sin(n) - \cos(n)$ (the sequence of values
of $\sin(x) - \cos(x)$ at the angles $x$ given by a whole number 
$n\ge 1$ in radian measure).  Show that there exists a subsequence 
$x_{n_k} = \sin(n_k) - \cos(n_k)$ that converges.  State any ``big theorems''
that 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 $f$ that is continuous at all real $c$, 
with $f(x) = 3$ for all $x \in {\bf Q}$, but $f(\sqrt{2}) = 4$.  
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\item{B)} (10)  A function $f$ that is continuous and bounded on an open 
interval $(a,b)$ with $M = {\rm lub} \{f(x) \mid x \in (a,b)\}$, but such that
$f(x) \ne M$ for all $x \in (a,b)$.  
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\item{C)} (10)  A continuous function on $[0,1]$ with $f(0) = 4$, $f(1) = -2$, 
but $f(x) \ne 0$ for all $x \in (0,1)$.  
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{\it Extra Credit.} (10) A function is said to be {\it right-continuous}
at $c$ if the one-sided limit $\lim_{x \to c^+} f(x)$ exists and equals $f(c)$.   
True or False (and give a proof or
a counterexample):  If $f$ is
right-continuous at all $c \in [a,b]$ for some closed interval $[a,b]$, then
$\{f(x) \mid x\in [a,b]\}$ is a bounded subset of ${\bf R}$.    
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