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\begin{document}

\title{Homework \#3, Real Analysis I, Fall 2012}


\author{Marius Ionescu}
\date{Due Friday, 09/24/2012}
\maketitle
\begin{problem}[Exercise 2.3.1 on page 49]
Show that the constant sequence $(a,a,a,a,\dots)$ 
converges to $a$.
\end{problem}
%\begin{proof}
  
%\end{proof}
\begin{problem}[Exercise 2.3.2 on page 49]
  Let $x_n\geq 0$ for all $n\in\N$
\begin{itemize}
\item[(a)] If $(x_n)\to 0$, show that $(\sqrt{x_n})\to 0$.
\item[(b)] If $(x_n)\to x$, show that $(\sqrt{x_n})\to \sqrt{x}$.
\end{itemize}
\end{problem}
%\begin{proof}
  
%\end{proof}
\begin{problem}[Exercise 2.3.3 on page 49, \textbf{Squeeze Theorem}]
Show that if 
$x_n\leq y_n\leq z_n$ for all $n\in\N$, and if 
$\lim x_n = \lim z_n = \ell$, then $\lim y_n=\ell$ as well.
\end{problem}
\begin{problem}[Exercise 2.3.6 on page 49]
  \begin{itemize}
\item[(a)] Show that if $(b_n)\to b$, then the sequence of absolute
values $|b_n|$ converges to $|b|$.
\item[(b)] Is the converse of part~(a) true?  If we know that 
$|b_n|\to|b|$, can we deduce that $(b_n)\to b$?
\end{itemize}
\end{problem}
\begin{problem}[Exercise 2.3.8 in page 50]
  Give an example of each of the following, or state that
  such a request is impossible by referencing the proper theorem(s).
  \begin{itemize}
  \item[(a)] sequences  $(x_n)$ and $(y_n)$, which both diverge,
    but whose sum $(x_n+y_n)$ converges;
  \item[(b)] sequences $(x_n)$ and $(y_n)$, whenr $(x_n)$ converges 
    $(y_n)$diverges, and $(x_n+y_n)$ converges;
  \item[(c)] a convergent sequence $(b_n)$ with $b_n\neq 0$ for all $n$
    such that $(1/b_n)$ diverges;
  \item[(d)] an unbounded sequence $(a_n)$ and a convergent sequence 
    $(b_n)$ with $a_n-b_n)$ bounded;
  \item[(e)] two sequences $(a_n)$ and $(b_n)$, where $(a_nb_n)$ and
      $a_n$ converge but $(b_n)$ does not.
  \end{itemize}
\end{problem}
\begin{problem}[Exercise 2.3.9 on page 50]
  Does Theorem~2.3.4 remain true if all of the inequalities
are assumed to be strict?  If we assume, for instance, that a convergent
sequence $(x_n)$ satisfies $x_n>0$ for all $n\in\N$, what may we conclude
about the limit?
\end{problem}
\end{document}
