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\newcommand{\myCourseNumber}{Math 142A}
\newcommand{\myAssignmentDueDate}{11:00pm Thursday, January 18, 2024}
\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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\begin{questions}

\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
\begin{parts}
\part Prove that \ $|a + b + c| \leq |a| + |b| + |c|$.
\part Use induction to prove that \ $|a_1 + a_2 + \cdots + a_n| \leq |a_1| + |a_2| + \cdots + |a_n|$  \ for any $n$ numbers $a_1, \, a_2, \, \ldots \, , \, a_n$.
\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 $S$ and $T$ be bounded subsets of $\mathbb{R}$.
\begin{parts}
\part Prove that if \ $S \subseteq T$, \ then \ $\inf(T) \leq \inf(S) \leq \sup(S) \leq \sup(T)$.
\part Prove that \ $\sup(S \cup T) = \max\{ \sup(S), \; \sup(T) \}$. \ Note: Be sure that you don't assume that \ $S \subseteq T$.
\end{parts}

\question Let $\mathbb{I}$ be set of real numbers that are not rational; that is, $\mathbb{I} = \mathbb{R} \setminus \mathbb{Q}$.  The elements of $\mathbb{I}$ are called \emph{irrational numbers}. Prove that if $a < b$, then there exists $x \in \mathbb{I}$ with $a < x < b$.

\question Prove that the following statements are equivalent:
\begin{parts}
\part $|a - b| < c$,
\part $b - c < a < b + c$,
\part $a \in (b - c, \; b + c)$.
\end{parts}

\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 $A$ and $B$ be nonempty bounded subsets of $\mathbb{R}$, and let $A + B$ be the set of all sums $a + b$ with $a \in A$ and $b \in B$; that is, $A + B = \left\{ a + b \mid a \in A \text{ and } b \in B \right\}$.
\begin{parts}
\part Prove that \ $\sup(A + B) = \sup(A) + \sup(B)$.
\part Prove that \ $\inf(A + B) = \inf(A) + \inf(B)$.
\end{parts}

\question Exhibit an example of:
\begin{parts}
\part a sequence \ $(x_n)$ \ of irrational numbers having a limit which is a rational number; that is, \ $(x_n) \subset \mathbb{I}$ \ with $\displaystyle \lim_{n \to \infty} x_n = r \in \mathbb{Q}$.
\part a sequence \ $(r_n)$ \ of rational numbers having a limit which is a irrational number; that is, \ $(r_n) \subset \mathbb{Q}$ \ with $\displaystyle \lim_{n \to \infty} r_n = x \in \mathbb{I}$.
\end{parts}

\end{questions}

\end{document}
