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\title{MATH3210 - Spring 2024 - Section 001}
\author{Homework 3}

\begin{document}

\maketitle

\begin{problem}[20 points]
Let $A,B \subset \R$ be nonempty subsets, and assume that if $x \in A$ or $x \in B$ then $x > 0$. Show that if $A/B = \set{ x/y : x \in A \mbox{ and }y \in B}$, then:

\[ \sup (A/B) = \frac{\sup A}{\inf B} \]

whenever $\inf B > 0$.
\end{problem}


\begin{problem}[30 points, 10 each]
Determine whether the sequence converges. If it converges, find its limit and prove that the sequence converges to that limit. If it diverges, prove that it diverges.

\begin{enumerate}[(a)]
\item $\set{\displaystyle\frac{2n+3}{8n+7}}$
\item $\set{\displaystyle\frac{n^2-100n}{7n+3}}$
\item  $\set{\sin(\pi \cdot n)}$
\end{enumerate} 
\end{problem}

\begin{problem}[Book 2.2.11, 20 points]
Let $(a_n)$ and $(b_n)$ be sequences, and assume that $b_n \to 0$ and $\abs{a_n} \le b_n$ for every $n \in \N$. Prove that $a_n \to 0$.
\end{problem}

\begin{problem}[10 points]
Show that if $I = [a,b]$ is a nonempty interval, and $x,y \in I$, then $\abs{x - y} \le b-a$.
\end{problem}

\begin{problem}[20 points]
Show that if $a_n \to L$ and $\abs{b_n - a_n} \to 0$, then $b_n \to L$.
\end{problem}

\end{document}


