\documentclass[letterpaper,12pt]{article}

\usepackage{amsfonts,amssymb,amsmath,amsthm,txfonts,pxfonts,mathrsfs,fullpage,enumitem}
\usepackage{xcolor}

\pagestyle{empty}
\begin{document} 

\begin{center}\textbf{Problem Set 3 \hspace{.7in} Real Analysis 1 \hspace{.7in} Due 9/15/14} \end{center}

\noindent Four of these problems will be graded, with each problem worth 5 points.  Clear and complete justification is required for full credit.  You are welcome to discuss these problems with anyone and everyone, but must write up your own final submission without reference to any sources other than the textbook and instructor. 

\begin{enumerate}
\item If a sequence $\{a_n\}$ converges to 0, and a sequence $\{b_n\}$ is bounded, then the sequence $\{a_n b_n\}$ converges to 0.

\item If a sequence $\{a_n\}$ converges to 0, and $\{a_n b_n\}$ converges to zero, then the sequence $\{b_n\}$ is bounded.

\item Determine whether $\lim_{n \rightarrow \infty} \frac{1}{n} \sin \frac{1}{n}$ exists, and find its value. [Kosmala 2.2.11(l)]

\item Consider the sequences $\{a_n\}$ and $\{b_n\}$, where sequence $\{a_n\}$ converges to zero.  Then $\{a_n b_n\}$ converges to zero. [Kosmala 2.2.12]

\item If a sequence $\{a_n\}$ diverges to $+\infty$ and $a_n \leq b_n$ for all $n \geq n_1$, then the sequence $\{b_n\}$ must also diverge to $+\infty$.

\item Do Exercise 2.3.4.

\item If sequences $\{a_n\}$ and $\{b_n\}$ diverge to $+\infty$, then $\{a_n b_n\}$ diverges to $+\infty$. [Kosmala 2.3.5]

\item Suppose that the sequence $\{a_n\}$ diverges to $+\infty$.  Find examples of sequences $\{a_n\}$ and $\{b_n\}$ so that $\left\{\frac{a_n}{b_n}\right\}$\\
	(a) diverges to $+\infty$.\\
	(b) converges to 7.\\
	(c) converges to 0.\\
	(d) diverges to $-\infty$.\\
	(e) oscillates.

\end{enumerate}

\end{document}