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\title{Math 3110: Homework 3}
%\author{Zach Norwood}
\date{Due Wednesday, 13 February 2019}

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\newtheorem {problem} {Problem}

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\begin{document}
\maketitle
\thispagestyle{empty}

\noindent
You will be graded on both the accuracy and the clarity of your solutions.
One purpose of the homework is to give you an opportunity to practice your proof-writing skills.

You are welcome\,---\,encouraged, even!\,---\,to collaborate on homework, but you should not 
copy solutions from any source, nor should you submit anything that you don't understand.

\begin{problem}
 Suppose that for each $n$ we have a closed interval $I_n = [a_n,b_n]$ and that the 
 intervals are decreasing: $I_0 \sups I_1 \sups I_2 \sups \cdots$.
 Prove that there is a real number that belongs to all of the intervals:
 $\bigcap_{n\in\N} I_n \ne \emptyset$.
\end{problem}

\begin{problem}
 Suppose that $X$ is a nonempty set.
 Prove that the following three assertions are equivalent:
 \begin{enumerate} [(a)]
  \item $X$ is finite or countably infinite.
  \item There is a one-to-one function $f\colon X\to\N$.
  \item There is an onto function $g\colon \N\to X$.
 \end{enumerate}
\end{problem}

\begin{problem}
This problem shows that ``equinumerosity is an equivalence relation.''
(This justifies the notation $\abs A = \abs B$.) Let $A$, $B$, and $C$ be sets.
For this problem only, we'll write $A\sim B$ to mean that $A$ and $B$ are equinumerous, meaning
that there is a bijection $A\to B$.
\begin{enumerate} [(a)]
 \item Show that $A \sim A$.
 \item Show that if $A \sim B$ then $B\sim A$.
 \item Show that if $A\sim B$ and $B\sim C$, then $A\sim C$.
\end{enumerate}
\end{problem}

\begin{problem}
 A set $I\subs \R$ is called an \textbf{interval} if for all triples of real numbers $x$, $y$, and $z$ such that
 $x < z < y$, if $x\in I$ and $y\in I$ then $z\in I$.
 \begin{enumerate} [(a)]
  \item Prove that every interval takes exactly one of the following nine forms: 
  \[
   (-\infty,a), \;
   (-\infty,a], \;
   (a,b), \;
   (a,b], \;
   [a,b), \;
   [a,b], \;
   (a,\infty), \;
   [a,\infty), \;\text{or }
   (-\infty,\infty).
   \]
  \item Give an example of an infinite family of intervals of the form $(a,b)$ of which any two are disjoint.
  \item Show that part (b) \emph{cannot} be solved with an uncountable family of intervals.
 \end{enumerate}
\end{problem}

\begin{problem}
   Suppose that $(a_n)_{n\in\N}$ and $(b_n)_{n\in\N}$ are sequences and that
  $\displaystyle\lim_{n\to\infty} a_n = L$ and $\displaystyle\lim_{n\to\infty} b_n = M \ne0$.
  Prove that $\displaystyle\lim_{n\to\infty} c_n = L/M$, where
  \[
   c_n = \begin{cases} \frac{a_n}{b_n} & \text{if } b_n \ne 0 \\
   				-3.17			& \text{if } b_n = 0.
		\end{cases}
  \]
  (\emph{Hint:} Start by showing that the $-3.17$ case occurs infrequently.)
\end{problem}

\begin{problem} 
 Suppose that $(a_n)_{n\in\N}$ is a sequence of nonnegative real numbers that converges to $L$.
  Show that the sequence $(\sqrt{a_n})_{n\in\N}$ converges to $\sqrt L$.
\end{problem}

\begin{problem}[The Sandwich Theorem for sequences] 
Suppose that $(l_n)_{n\in\N}$, $(u_n)_{n\in\N}$, and $(c_n)_{n\in\N}$ 
  are all sequences such that $l_n \le c_n \le u_n$ for every $n\in\N$. Suppose also that
  $\displaystyle\lim_{n\to\infty} l_n = \displaystyle\lim_{n\to\infty} u_n = L$. Prove that $(c_n)_{n\in\N}$ also converges to $L$.\footnote{%
  $l$ is for \emph{lower} and $u$ is for \emph{upper}.}
\end{problem}

\end{document}




































