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

% set 1-inch margins in the document
\usepackage[margin=1in]{geometry}

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\usepackage{graphicx}

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\name{}
\class{Math 315}
\assignment{Homework \#4}
\duedate{4/18/2017}

% 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: 2.2.2, 2.2.3, 2.2.5, 2.2.6}

\begin{problem}[2.2.2]
Verify, using the definition of convergence of a sequence, that the following sequences converge to the proposed limit.
\begin{enumerate}
\item $\lim \frac{2n+1}{5n+4} = \frac{2}{5}$.
\item $\lim \frac{2n^2}{n^3 + 3} = 0$.
\item $\lim \frac{\sin(n^2)}{\sqrt[3]{n}} = 0$.
\end{enumerate}
\end{problem}

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

\newpage

\begin{problem}[2.2.3]
Describe what we would have to demonstrate in order to disprove each of the following statements.
\begin{enumerate}
\item At every college in the United States, there is a student who is at least seven feet tall.
\item For all colleges in the United States, there exists a professor who gives every student a grade of either A or B.
\item There exists a college in the United States where every student is at least six feet tall.
\end{enumerate}
\end{problem}

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

\newpage

\begin{problem}[2.2.5]
Let $[[x]]$ be the greatest integer less than or equal to $x$. For example, $[[\pi]] = 3$ and $[[3]] = 3$. For each sequence, find $\lim a_n$ and verify it with the definition of convergence.
\begin{enumerate}
\item $a_n = [[5/n]]$,
\item $a_n = [[(12 + 4n)/3n]]$.
\end{enumerate}
Reflecting on these examples, comment on the statement following Definition 2.2.3 that ``the smaller the $\epsilon$-neighborhood, the larger $N$ may have to be.''
\end{problem}

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

\newpage

\begin{problem}[2.2.6]
Prove Theorem 2.2.7. To get started, assume $(a_n) \to a$ and also that $(a_n) \to b$. Now argue $a = b$.
\end{problem}

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

\newpage

\end{document}
