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	 \author{Replace this text with your name\vspace{-1ex}}
	 \date{Due: Replace this text with a due date}
	 \title{\textbf{\large{Real Analysis Homework \#2}}\vspace{-1ex}}
	 
	 \begin{document}
	 \maketitle
	 
	 \begin{exercise}[2.2.2]
	 Verify, using the definition of convergence of a sequence, that the following sequences converge to the proposed limit.
	 \begin{enumerate}[label=(\alph*)]
	 	\item $\lim\frac{2n+1}{5n+4}=\frac{2}{5}$.
	 	\item $\lim\frac{2n^2}{n^3+3}=0$.
	 	\item $\lim\frac{\sin(n^2)}{\sqrt[3]{n}}=0$.
	 \end{enumerate}
	 \end{exercise}
	 
	 \begin{solution}
	 Replace this text with your solution.
	 \end{solution}
	 
	 \begin{exercise}[2.2.5]
	 Let $[[x]]$ be the greatest integer less than or equal to $x$. For example, $[[\pi]]=3$ and $[[3]]=3$. For each sequence, find $\lim a_n$ and verify it with the definition of convergence.
	 \begin{enumerate}[label=(\alph*)]
	 	\item $a_n=[[5/n]]$,
	 	\item $a_n=[[(12+4n)/3n]]$.
	 \end{enumerate}
	 \end{exercise}
	 
	 \begin{solution}
	 Replace this text with your solution.
	 \end{solution}
	 
	 \newpage
	 
	 \begin{exercise}[2.3.1]
	 Let $x_n\geq 0$ for all $n\in\mathbf{N}$.
	 \begin{enumerate}[label=(\alph*)]
	 	\item If $(x_n)\to 0$, show that $(\sqrt{x_n})\to 0$.
	 	\item If $(x_n)\to x$, show that $(\sqrt{x_n})\to \sqrt{x}$.
	 \end{enumerate}
	 \end{exercise}
	 
	 \begin{solution}
	 Replace this text with your solution.
	 \end{solution}
	 
	 \begin{exercise}[2.3.3]\textbf{(Squeeze Theorem).}
	 Show that if $x_n\leq y_n\leq z_n$ for all $n\in\mathbf{N}$, and if $\lim x_n=\lim z_n=l$, then $\lim y_n=l$ as well.
	 \end{exercise}
	 
	 \begin{solution}
	 Replace this text with your solution.
	 \end{solution}
	 
	 \newpage
	 
	 \begin{exercise}[2.4.1]
	 \begin{enumerate}[label=(\alph*)]
	 	\item Prove that the sequence defined by $x_1=3$ and
	 	\[
	 	x_{n+1}=\frac{1}{4-x_n}
	 	\]
	 	converges.
	 	\item Now that we know $\lim x_n$ exists, explain why $\lim x_{n+1}$ must also exist and equal the same value.
	 	\item Take the limit of each side of the recursive equation in part (a) to explicitly compute $\lim x_n$.
	 \end{enumerate}
	 \end{exercise}
	 
	 \begin{solution}
	 Replace this text with your solution.
	 \end{solution}
	 
	 \begin{exercise}[2.4.3]
	 \begin{enumerate}[label=(\alph*)]
	 	\item Show that
	 	\[
	 	\sqrt{2},\sqrt{2+\sqrt{2}},\sqrt{2+\sqrt{2+\sqrt{2}}},\ldots
	 	\]
	 	converges and find the limit.
	 	\item Does the sequence
	 	\[
	 	\sqrt{2},\sqrt{2\sqrt{2}},\sqrt{2\sqrt{2\sqrt{2}}},\ldots
	 	\]
	 	converge? If so, find the limit.
	 \end{enumerate}
	 \end{exercise}
	 
	 \begin{solution}
	 Replace this text with your solution.
	 \end{solution}
	 
	 \newpage
	 
	 \begin{exercise}[2.5.3]
	 \begin{enumerate}[label=(\alph*)]
	 	\item Prove that if an infinite series converges, then the associative property holds. Assume $a_1+a_2+a_3+a_4+a_5+\cdots$ converges to a limit $L$ (i.e., the sequence of partial sums $(s_n)\to L$). Show that any regrouping of the terms
	 	\[
	 	(a_1+a_2+\cdots+a_{n_1})+(a_{n_1+1}+\cdots+a_{n_2})+(a_{n_2+1}+\cdots+a_{n_3})+\cdots
	 	\]
	 	leads to a series that also converges to $L$.
	 	\item Compare this result to the example discussed at the end of Section 2.1 where infinite addition was shown not to be associative. Why doesn't our proof in (a) apply to this example?
	 \end{enumerate}
	 \end{exercise}
	 
	 \begin{solution}
	 Replace this text with your solution.
	 \end{solution}
	 
	 \begin{exercise}[2.5.5]
	 Assume $(a_n)$ is a bounded sequence with the property that every convergent subsequence of $(a_n)$ converges to the same limit $a\in\mathbf{R}$. Show that $(a_n)$ must converge to $a$.
	 \end{exercise}
	 
	 \begin{solution}
	 Replace this text with your solution.
	 \end{solution}
	 
	 \newpage
	 
	 \begin{exercise}[2.6.2]
	 Give an example of each of the following, or argue that such a request is impossible.
	 \begin{enumerate}[label=(\alph*)]
	 	\item A Cauchy sequence that is not monotone.
	 	\item A Cauchy sequence with an unbounded subsequence.
	 	\item A divergent monotone sequence with a Cauchy subsequence.
	 	\item An unbounded sequence containing a subsequence that is Cauchy.
	 \end{enumerate}
	 \end{exercise}
	 
	 \begin{solution}
	 Replace this text with your solution.
	 \end{solution}
	 
	 \begin{exercise}[2.6.4]
	 Let $(a_n)$ and $(b_n)$ be Cauchy sequences. Decide whether each of the following sequences is a Cauchy sequence, justifying each conclusion.
	 \begin{enumerate}[label=(\alph*)]
	 	\item $c_n=|a_n-b_n|$
	 	\item $c_n=(-1)^na_n$
	 	\item $c_n=[[a_n]]$, where $[[x]]$ refers to the greatest integer less than or equal to $x$.
	 \end{enumerate}
	 \end{exercise}
	 
	 \begin{solution}
	 Replace this text with your solution.
	 \end{solution}
	 
	 \newpage
	 
	 \begin{exercise}[2.7.7]
	 \begin{enumerate}[label=(\alph*)]
	 	\item Show that if $a_n>0$ and $\lim(na_n)=l$ with $l\ne 0$, then the series $\sum a_n$ diverges.
	 	\item Assume $a_n>0$ and $\lim (n^2a_n)$ exists. Show that $\sum a_n$ converges.
	 \end{enumerate}
	 \end{exercise}
	 
	 \begin{solution}
	 Replace this text with your solution.
	 \end{solution}
	 
	 \begin{exercise}[2.7.9](Ratio Test). Given a series $\sum_{n=1}^\infty a_n$ with $a_n\ne 0$, the Ratio Test states that if $(a_n)$ satisfies
	 \[
	 \lim\left|\frac{a_{n+1}}{a_n}\right|=r<1,
	 \]
	 then the series converges absolutely.
	 \begin{enumerate}[label=(\alph*)]
	 	\item Let $r'$ satisfy $r<r'<1$. Explain why there exists an $N$ such that $n\geq N$ implies $|a_{n+1}|\leq |a_n|r'$.
	 	\item Why does $|a_N|\sum (r')^n$ converge?
	 	\item Now, show that $\sum |a_n|$ converges, and conclude that $\sum a_n$ converges. 
	 \end{enumerate}
	 \end{exercise}
	 
	 \begin{solution}
	 Replace this text with your solution.
	 \end{solution}
	 
	 \end{document}