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\newcommand{\myCourseNumber}{Math 142A}
\newcommand{\myAssignmentDueDate}{11:00pm Friday, October 7, 2022}
\newcommand{\myAssignmentTitle}{Homework Assignment 1}
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\begin{document}
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\firstpageheader{\bfseries \myCourseNumber \enspace \myAssignmentTitle \\ Due \myAssignmentDueDate }{}{}
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%\firstpagefooter{}{}{(This exam is worth {\numpoints} points.)}
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\begin{questions}

\question Show that \enspace $\sqrt{4 + 2 \sqrt{3}} - \sqrt{3}$ \enspace is a rational number.

\question Find all rational solutions of the equation \enspace $x^8 - 4 x^5 +13 x^3 - 7 x + 1 = 0$.  Be sure to explain how you know you found all the rational solutions.

\question
\begin{parts}
\part Show \enspace $|b| \leq a$ \enspace if and only if \enspace $-a \leq b \leq a$.
\part Prove \enspace $\left| \, \left| a \right| - \left| b \right| \, \right| \leq |a - b|$ \enspace for all $a, b \in \mathbb{R}$.
\end{parts}

\question Let $a, b \in \mathbb{R}$. Show that if \enspace $a < b_1$ \enspace for every \enspace $b_1 > b$, \enspace then \enspace $a \leq b$.

\question Prove that if \enspace $a > 0$, \enspace then there exists $n \in \mathbb{N}$ such that \enspace $\displaystyle \frac{1}{n} < a < n$.

\question Let $a, b \in \mathbb{R}$.  Show that if \enspace $a \leq b + \frac{1}{n}$ \enspace for all $n \in \mathbb{N}$, then \enspace $a \leq b$.

\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 Let $(s_n)$ be a sequence that converges.
\begin{parts}
\part Show that if \enspace $s_n \geq a$ \enspace for all but finitely many $n$, then \enspace $\lim s_n \geq a$.
\part Show that if \enspace $s_n \leq b$ \enspace for all but finitely many $n$, then \enspace $\lim s_n \leq b$.
\part Conclude that if all but finitely many $s_n$ belong to $[a,b]$, then $\lim s_n$ belongs to $[a,b]$.
\end{parts}

\end{questions}

\end{document}
