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\title{MATH3210 - Spring 2024 - Section 001}
\author{Homework 4}

\begin{document}

\maketitle

\begin{problem}[20 points]
Let $(c_n)$ denote any sequence of real numbers such that $0 < c_n < 1$. Show that if $(a_n)$ is a sequence defined recursively by $a_1 = 1$ and $a_{n+1} = c_n \cdot a_n$, then $(a_n)$ converges. [{\it Hint}: You won't be able to use the definition of a limit directly, as some of these sequences won't converge to 0! What tools do we have for proving a limit exists without knowing what the limit is?]
\end{problem}



\begin{problem}[40 points]
Let $(a_n)$ be a sequence of real numbers such that $a_n >  0$ for every $n \in \N$. For each, prove or find a counterexample:

\begin{enumerate}[(a)]
\item If $a_n$ diverges to $\infty$, then $ 1/a_n \to 0$.
\item If $1/a_n \to 0$, then $(a_n)$ is unbounded.
\item If $(a_n)$ is bounded above, then $(1/a_n)$ has a convergent subsequence.
\item If $\liminf a_n > 0$, then $(1/a_n)$ has a convergent subsequence.
\end{enumerate}
\end{problem}

For the next two exercises, we consider the {\it asymptotic variation} of a sequence $(a_n)$. If $(a_n)$ is a sequence, let $V = \limsup a_n - \liminf a_n$. We assume throughout that $V < \infty$.

\begin{problem}[20 points]
 Show that for any $\ve > 0$ and $N \in \N$, there exist indices $m,n \in \N$ such that $m,n \ge N$ and $a_m - a_n > V - \ve$.
\end{problem}

\begin{problem}[20 points]
Give an example of a sequence such that for any two indices $m,n \in \N$, $a_m - a_n < V$. Justify your answer (ie, calculate $V$ for your sequence and prove that your calculation is correct. Then verify the given property).
\end{problem}

\begin{problem}[Ungraded, extra practice]
Show thta for any $\ve > 0$, there exists some $N \in \N$ such that if $m,n \in \N$ satisfy $m,n \ge N$, $\abs{a_m - a_n} < V +\ve$.
\end{problem}

\end{document}


