% HMC Math dept HW template example
% v0.04 by Eric J. Malm, 10 Mar 2005
\documentclass[12pt,letterpaper,boxed]{hmcpset}

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\usepackage[margin=1in]{geometry}

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\usepackage{graphicx}

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\renewcommand{\labelenumi}{(\alph{enumi})}

% info for header block in upper right hand corner
\name{}
\class{Math 315}
\assignment{Homework \#2}
\duedate{4/11/2017}

% macros
\newcommand{\NN}{\mathbb{N}}
\newcommand{\ZZ}{\mathbb{Z}}
\newcommand{\QQ}{\mathbb{Q}}
\newcommand{\RR}{\mathbb{R}}
\newcommand{\II}{\mathbb{I}}

\begin{document}

\problemlist{Abbott: 1.3.5, 1.3.8, 1.4.1, 1.4.3}

\begin{problem}[1.3.5]
As in Example 1.3.7, let $A \subseteq \RR$ be nonempty and bounded above, and let $c \in \RR$. This time, define the set $cA$ by $cA = \{ca : a \in A\}$.
\begin{enumerate}
\item If $c \geq 0$, show that $\sup(cA) = c \sup A$.
\item Postulate a similar type of statement for $\sup(cA)$ for the case $c < 0$.
\end{enumerate}
\end{problem}

%\begin{solution}
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%\end{solution}

\newpage

\begin{problem}[1.3.8]
Compute, without proofs, the suprema and infima of the following sets:
\begin{enumerate}
\item $\{m/n : m, n \in \NN \ \text{with} \ m < n \}$.
\item $\{(-1)^m/n : m, n \in \NN\}$.
\item $\{n/(3n + 1) : n \in \NN\}.$
\item $\{m/(m + n) : m,n \in \NN\}$.
\end{enumerate}
\end{problem}

%\begin{solution}
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%\end{solution}

\newpage

\begin{problem}[1.4.1]
Recall that $\II$ stands for the set of irrational numbers.
\begin{enumerate}
\item Show that if $a, b \in \QQ$, then $ab$ and $a+b$ are elements of $\QQ$ as well.
\item Show that if $a \in \QQ$ and $t \in \II$, then $a+t \in \II$ and $at \in \II$ as long as $a \neq 0$.
\item Part (a) can be summarized by saying that $\QQ$ is closed under addition and multiplication. Is $\II$ closed under addition and multiplication? Given two irrational numbers $s$ and $t$, what can we say about $s+t$ and $st$?
\end{enumerate}
\end{problem}

%\begin{solution}
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%\end{solution}

\newpage

\begin{problem}[1.4.3]
Prove that $\bigcap_{n=1}^{\infty} (0, 1/n) = \emptyset$. Notice that this demonstrates that the intervals in the Nested Interval Property must be closed for the conclusion of the theorem to hold.
\end{problem}

%\begin{solution}
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%\end{solution}

\newpage

\end{document}
