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\newcommand{\myCourseNumber}{Math 142A}
\newcommand{\myAssignmentDueDate}{11:00pm Tuesday, February 10, 2025}
\newcommand{\myAssignmentTitle}{Homework Assignment 3}
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\begin{document}

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\begin{questions}
    
\question
\begin{parts}
\part Let $(s_n)$ be a monotone sequence.  Prove that $(s_n)$ converges if and only if $(s_n^2)$ converges.
\part Find a non-monotone sequence $(t_n)$ such that $(t_n^2)$ converges but $(t_n)$ does not converge.
\end{parts}

\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 $\lim\limits_{k \to \infty} r_{n_k} = +\infty$.

\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 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 $(s_n)$ and $(t_n)$ be bounded sequences of nonnegative real numbers.  Prove that
\[ \lim\sup s_n \, t_n \leq \left( \lim\sup s_n \right) \left( \lim\sup t_n \right). \]

\question Prove the following theorem.
\begin{theorem}
Let $(s_n)$ be a bounded sequence. \ $\lim \inf s_n = \underline{s}$ \ if and only if for every $\varepsilon > 0$,
\begin{enumerate}
\item[(i)] $s_n > \underline{s} - \varepsilon$ for all but finitely many $n$, and
\item[(ii)] $s_n < \underline{s} + \varepsilon$ for infinitely many $n$.
\end{enumerate}
\end{theorem}


\question Let $(s_n)$ be a bounded sequence of nonzero real numbers.  Prove that
\[ \lim\inf \left| \frac{s_{n+1}}{s_n} \right|  \leq \lim\inf \left| s_n \right|^{1/n} . \]

\question Let $(s_n)$ be a sequence of nonnegative numbers.  For each $n$, define $\displaystyle \sigma_n = \frac{1}{n} \left(s_1 + s_2 + \cdots + s_n \right)$.  $(\sigma_n)$ is called the sequence of Ces\`{a}ro means for $(s_n)$.
\begin{parts}
\part Show that \enspace $\displaystyle \lim\inf s_n \leq \lim\inf \sigma_n \leq \lim\sup \sigma_n \leq \lim\sup s_n$.
\part Show that if \ $\lim s_n$ \ exists, then \ $\lim \sigma_n$ \ exists and \ $\lim \sigma_n = \lim s_n$.
\part Exhibit an example for which \ $\lim \sigma_n$ \ exists, but \ $\lim s_n$ \ does not exist.
\end{parts}

\end{questions}

\end{document}