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\begin{document}

\title{Homework \#4, Real Analysis I, Fall 2012}


\author{Marius Ionescu}
\date{\bf{Due Monday, 10/01/2012; Note that it is due on Monday
    instead of Friday!}}
\maketitle
\begin{problem}[Exercise 2.4.2 on page 54]
\begin{enumerate}
\item[(a)] Prove that the sequence defined by $x_1 = 3$ and 
\[
x_{n+1} = \frac1{4-x_n}
\]
converges.
\item[(b)] Now that we know $\lim x_n$ exists, explain why $\lim x_{n+1}$ 
must also exist and equal the same value.
\item[(c)] Take the limit of each side of the recursive equation in part~(a)
of this exercise to explicitly compute $\lim x_n$.
\end{enumerate}
\end{problem}
%\begin{proof}
  
%\end{proof}
\begin{problem}[Exercise 2.4.4 on page 54]
Show that 
\[
\sqrt2,\ \sqrt{2\sqrt2},\ \sqrt{2\sqrt{2\sqrt2}},\ \dots
\]
converges and find the limit.
\end{problem}
%\begin{proof}
  
%\end{proof}
\begin{problem}[Exercise 2.4.5 on page 54, \textbf{(Calculating Square Roots).}] Let $x_1 = 2$, and 
define
\[
x_{n+1} = \frac12\left(x_n + \frac2{x_n}\right)\ .
\]

\begin{itemize}
\item[(a)] Show that $x_n^2$ is always greater than 2, and then use this
to prove that $x_n-x_{n+1}\geq 2$.  Conclude that $\lim x_n = \sqrt 2$.
\item[(b)] Modify the sequence $(x_n)$ so that it converges to
  $\sqrt{c}$.
\end{itemize}
\end{problem}
\begin{problem}[Exercise 2.5.3 on page 58]Give an example of each of the following, or argue that 
such a request is impossible
\begin{itemize}
\item[(a)] A sequence that does not contain 0 or 1 as a term but contains 
subsequences converging to each of these values.
\item[(b)] A monotone sequence that diverges but has a convergent 
subsequence.
\item[(c)] A sequence that contains subsequences converging to every 
point in the infinite set $\{1,1/2,1/3,1/4,1/5,\dots\}$.
\item[(d)] An unbounded sequence with a convergent subsequence.
\item[(e)] A sequence that has a subsequence that is bounded but contains
no subsequence that converges.
\end{itemize}
\end{problem}
\begin{problem}[Exercise 2.6.1 on page 61]
  Give an example of each of the following, or argue
that such a request is impossible.
\begin{itemize}
\item[(a)] A Cauchy sequence that is not monotone.
\item[(b)] A monotone sequence that is not Cauchy. 
\item[(c)] A Cauchy sequence with a divergent subsequence.
\item[(d)] An unbounded sequence containing a subsequence that 
is Cauchy.
\end{itemize}
\end{problem}
\begin{problem}[Exercise 2.6.3 on page 61]
  \begin{itemize}
\item[(a)] Explain how the following pseudo-Cauchy property 
differs from the proper definition of a Cauchy sequence: A sequence 
$(a_n)$ is \textit{pseudo-Cauchy} if, for all $\varepsilon>0$, there 
exists an $N$ such that if $n\geq N$, then 
$|s_{n+1} - s_n|<\varepsilon$.
\item[(b)] If possible, give an example of a divergent sequence 
$(s_n)$ that is pseudo-Cauchy.
\end{itemize}
\end{problem}
\end{document}
