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\noindent
\parbox{2.25in}{\bf Math 2110-410}
\hfill {\Large \bf Homework \#4} \hfill
\parbox{2.25in}{\bf \hfill Due: Fri., Feb. $28^\mathrm{th}$, 2020}

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\begin{enumerate}
\item Prove that $ \displaystyle{ \left\langle \frac{1}{n^3+12345} \right\rangle }$ converges to 0.
\item Show that $\displaystyle{\left\langle \frac{n^2+5}{4n^2} \right\rangle}$ where $ n > 0 $ converges to $\displaystyle{\frac{1}{4} }$.
\item Consider the sequence $\displaystyle{ \left\langle (-1)^n n^2 \right\rangle }$. This sequence obviously diverges.
   \begin{enumerate}[(a)]
   \item Prove that this sequence diverges by showing it is unbounded.
   \item Prove that this sequence diverges by showing it does not converge to any real number $ L $.
   \end{enumerate}
\item Suppose that $ \displaystyle{ \left\langle a_n \right\rangle } $ converges to $ L $. Show that $ \displaystyle{ \left\langle |a_n| \right\rangle } $ converges to $ |L| $.
\item Suppose that $ \displaystyle{ \left\langle a_n \right\rangle } $ and $ \displaystyle{ \left\langle b_n \right\rangle } $ converge to 0. Show that $ \displaystyle{ \left\langle a_nb_n \right\rangle } $ converges to 0 as well.
\end{enumerate}

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