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\newcommand{\myCourseNumber}{Math 142A}
\newcommand{\myAssignmentDueDate}{Tuesday, January 30, 2024}
\newcommand{\myAssignmentTitle}{Homework Assignment 2}
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\begin{questions}

\question Let $(t_n)$ be a bounded sequence; that is, there exists $M \geq 0$ such that \enspace $|t_n| \leq M$ \enspace for all $n$.  Let $(s_n)$ be a sequence such that \enspace $\lim s_n = 0$. \enspace Prove that \enspace $\lim (s_n t_n) = 0$.

\question Consider three sequences $(a_n), (b_n)$, and $(s_n)$ such that \enspace $a_n \leq s_n \leq b_n$ \enspace for all $n$, and \newline $\lim a_n = \lim b_n = s$. \enspace Prove that \enspace $\lim s_n = s$.

\question Suppose $(s_n)$ and $(t_n)$ are sequences such that \enspace $|s_n| \leq t_n$ \enspace for all $n$ and \enspace $\lim t_n = 0$. \enspace Prove that $\lim s_n = 0$.

\question Suppose that $s_n \not= 0$ for every index $n$ and that the limit \enspace $L = \lim \left\vert \frac{s_{n+1}}{s_n} \right\vert$ \enspace is defined.
\begin{parts}
\part Show that if \enspace $L < 1$, \enspace then \enspace $\lim s_n = 0$.
\part Show that if \enspace $L > 1$, \enspace then \enspace $\lim \left\vert s_n \right\vert = +\infty$.
\end{parts}
(See Exercise 9.12 in your text for a hint.)

\question Let \enspace $s_1 = 1$, \enspace and for \enspace $n \geq 1$, \enspace let \enspace $s_{n+1} = \sqrt{s_n + 1}$. \enspace It turns out that $(s_n)$ converges.  Assume this fact and show that limit \enspace $\lim s_n = \frac{1}{2}\left(1 + \sqrt{5}\right)$.

\question Let \enspace $x_1 = 1$ \enspace and \enspace $x_{n+1} = 3 x_n^2$ \enspace for $n \geq 1$.
\begin{parts}
\part Show that if \enspace $a = \lim x_n$, \enspace then \enspace $a = \frac{1}{3}$ \enspace or \enspace $a = 0$.
\part Does \enspace $\lim x_n$ \enspace exist? Justify your answer.
\part Explain the apparent contradiction between the result in part (a) and part (b).
\end{parts}

\question Show that \enspace $\displaystyle \lim_{n \to \infty} \frac{a^n}{n !} = 0$ \enspace for all \enspace $a \in \mathbb{R}$.

\question
\begin{parts}
\part Verify that \enspace $\displaystyle 1 + a + a^2 + \cdots + a^n = \frac{1 - a^{n+1}}{1-a}$ \enspace for \enspace $a \not= 1$.
\part Determine \enspace $\displaystyle \lim_{n \to \infty} \left(1 + a + a^2 + \cdots + a^n\right)$ \enspace for \enspace $|a| < 1$.
\part What is \enspace $\displaystyle \lim_{n \to \infty} \left(1 + a + a^2 + \cdots + a^n\right)$ \enspace for \enspace $a \geq 1$?
\end{parts}

\question Let $S$ be a bounded nonempty subset of $\mathbb{R}$ such that \enspace $\sup(S) \not\in S$. \enspace Show that there is a sequence $(s_n)$ of points in $S$ such that \enspace $\lim s_n = \sup(S)$.

\question Let $(s_n)$ be a sequence such that \enspace $\displaystyle \left|s_{n+1} - s_n\right| < 2^{-n} \text{ for all } n \in \mathbb{N}$.
\begin{parts}
\part Prove that $(s_n)$ is a Cauchy sequence and, therefore, a convergent sequence.
\part Is $(s_n)$ a Cauchy sequence if we only assume that \enspace $\displaystyle \left|s_{n+1} - s_n\right| < \frac{1}{n} \text{ for all } n \in \mathbb{N}$?
\end{parts}

\end{questions}

\end{document}
