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\centerline{MATH 242 -- Principles of Analysis}
\centerline{Problem Set 3 -- due:  Feb. 15}
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\noindent
{\it `A` Section}
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\item{1.}  A set $B$ is said to be {\it finite} if there is some $n \in {\bf N}$ (the number of
elements in $B$), and a one-to-one onto mapping $f : \{1,2,\ldots, n\} \to B$.  (Intuitively, we
think that $f(1) = b_1, f(2) = b_2, \ldots$ ``counts through'' all the elements of $B$ one at 
a time without repetitions and without missing any elements in $B$.)
For each of the following sets, either show $B$ is finite by determining the $n$ and constructing a mapping
$f$ as above, or say why no such mapping exists.
\itemitem{a.} $B = \{r = p/q \in {\bf Q} \mid 1 \le q \le 5\ {\rm and}\ 0 < r < 1\}$
\itemitem{b.} $B = \{r = p/q \in {\bf Q} \mid 0 < r < 1\}$
\itemitem{c.} $B = \{n \in {\bf Z} \mid |n| \le 10^9\}$
\item{2.}  Which of the following sequences converge to 0?  Explain your answers.
\itemitem{a.}  $\{x_n\}$, where
$$x_n = \cases{e^n & if $n \le 1000$\cr
               e^{-n} & if $n > 1000$\cr}$$
\itemitem{b.}  $\{y_n\}$, where
$$y_n = \cases{1 & if $n$ is evenly divisible by $1000$ \cr
               {1\over n} & if $n$ is not evenly divisible by $1000$\cr}$$
\itemitem{c.}  $\{z_n\}$, where
$$z_n = \cases{n & if $n$ is a Fermat prime number (look up on Wikipedia)\cr
               {(-1)^n\over n^2} & if $n$ is not a Fermat prime number\cr}$$
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\item{3.}  Let $f(x) = [x]$ be the greatest integer function, defined as $[x] = $ the greatest integer $\le x$.
\itemitem{a.}  If $x_n \to a$, does it follow that $[x_n] \to [a]$?  Prove or give a counterexample.
\itemitem{b.}  If $[x_n] \to [a]$, does it follow that $x_n \to a$?  Prove or give a counterexample.
\item{4.}
\itemitem{a.}  Suppose that $x_n \to e$ (the base of the natural logarithms).  Explain why there exists
an $n_0$ such that $x_n < 3$ for all $n \ge n_0$.
\itemitem{b.}  Suppose $x_n \to 6$ and $y_n \to 9$.  Explain why there exists an $n_0$ such that
$x_n + y_n > 14$ for all $n \ge n_0$.
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\noindent
{\it `B` Section}
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\item{1.}
\itemitem{a.}  Prove that $\sqrt{5}$ is an irrational number.
\itemitem{b.}  If $r \ne 0$ and $s$ are rational numbers, show that $r\sqrt{5} + s$ is also
an irrational number.
\itemitem{c.} If $x = r\sqrt{5} + s$ and $x' = r'\sqrt{5} + s'$ are two numbers as in part b, what
can be said about $x + x'$ and $xx'$?  Are they necessarily irrational too?  
\item{2.}  Let $A$ and $B$ be two nonempty sets of real numbers. 
\itemitem{a.} Assume that $x \le y$ for all $x \in A$
and $y\in B$. Show that ${\rm lub}\ A$ and ${\rm glb}\ B$ must exist.
\itemitem{b.}  Under the same assumptions as part a, show that ${\rm lub}\ A \le {\rm glb}\ B$.
\itemitem{c.}  Now assume that $A$ and $B$ are bounded.  
Is it true that ${\rm lub}\ A \le {\rm glb}\ B$ implies that $x \le y$ for all $x \in A$
and $y\in B$?  Prove or give a counterexample.
\item{3.}  Let $A$ be a bounded set of real numbers and let $B = \{kx\mid x \in A\}$, where
$k < 0$ is a strictly negative number.  Show that $B$ is also bounded.  Then, determine formulas
for computing  ${\rm lub}\ B$
and ${\rm glb}\ B$ in terms of ${\rm lub}\ A$ and ${\rm glb}\ A$, and prove your assertions.
\item{4.}  Determine whether each of the following sequences converge and prove your assertions using 
the $\varepsilon, n_0$ definition of convergence.
\itemitem{a.}  $x_n = {3n^2\over n^2 + 5}$
\itemitem{b.}  $x_n = {1\over \ln(n)}$
\itemitem{c.}  $x_n = \cases{{3n + 1\over 4n} & if $n$ is even\cr
                             {6n - 3\over 8n + 1} & if $n$ is odd\cr}$
\itemitem{d.}  $x_n = \sin(n\pi/2)$.  
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