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\newcommand{\myCourseNumber}{Math 142A}
\newcommand{\myAssignmentDueDate}{11:00pm Friday, October 21, 2022}
\newcommand{\myAssignmentTitle}{Homework Assignment 3}
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\begin{document}

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\begin{questions}
    
\question Let $(r_n)$ be an enumeration of the set $\mathbb{Q}$ of all rational numbers.  Show there exists a subsequence $(r_{n_k})$ such that \enspace $\lim_{k \to \infty} = +\infty$.

\question Prove that \enspace $\lim \inf s_n = - \lim \sup (-s_n)$ \enspace for every sequence $(s_n)$. \newline
[Hint: See Definition 10.6 and Exercise 5.4]

\question Let $S$ be a bounded set.  Prove that there is an increasing sequence $(s_n)$ in $S$ such that \enspace $\lim s_n = \sup S$. \enspace Explain why if \enspace $\sup S$ \enspace is in $S$, then it suffices to define $s_n = \sup S$ for all $n$.

\question Let $(s_n)$ be an increasing sequence.  Show that $(s_n)$ has no dominant terms.

\question Let $(s_n)$ be a strictly decreasing sequence; that is, $s_n > s_{n+1}$ for every $n \in \mathbb{N}$.  Show that every term of $(s_n)$ is a dominant term.

\question Show that a monotonically increasing sequence is bounded if it has a bounded subsequence.

\question Suppose the sequence $(s_n)$ is monotonically increasing and that it has a convergent subsequence.  Show that $(s_n)$ converges.

\question Let $c > 0$.  Consider the quadratic equation \enspace $\displaystyle x^2 - x - c = 0$, \enspace where $x > 0$.  Define the sequence $(x_n)$ recursively, as follows:
\begin{displaymath}
x_n = \begin{cases}
1 & \text{if } n = 1 \\
\sqrt{c + x_{n-1}} & \text{if } n > 1
\end{cases}
\end{displaymath}
Prove that $(x_n)$ converges monotonically to the solution of the above quadratic equation.

\end{questions}

\end{document}
