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\begin{center}\textbf{Problem Set 3 \hspace{.7in} Real Analysis 1 \hspace{.7in} Due 9/16/24} \end{center}

\noindent Ten of these problems will be graded, with each problem worth 2 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 Show that $\left \{ \frac{b-a}{2^n} \right \}$ converges to 0.

\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)]\\

For 5-9, suppose that the sequence $\{a_n\}$ diverges to $+\infty$.  Find examples of sequences $\{a_n\}$ and $\{b_n\}$ so that $\left\{\frac{a_n}{b_n}\right\}$:
\item diverges to $+\infty$.
\item converges to 7.
\item converges to 0.
\item diverges to $-\infty$.
\item oscillates.\\

\item Suppose $\{a_n\}$ and $\{b_n\}$ both oscillate.  Then $\{a_n \cdot b_n \}$ and $\{a_n + b_n \}$ oscillate.

\item If $\{a_n\}$ is a Cauchy sequence and $S=\{a_n|n \in \mathbb N \}$ is finite, then $\{a_n\}$ is constant from some point on.

\item Let $s_0$ be an accumulation point of $S$.  Then any neighborhood of $s_0$ contains at least one point of $S$ different from $s_0$ iff any neighborhood of $s_0$ contains infinitely many points of $S$.

\end{enumerate}

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