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\title{Math 3110: Homework 2}
%\author{Zach Norwood}
\date{Due Wednesday, 6 February 2019}

\theoremstyle {definition}
\newtheorem {problem} {Problem}

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\newcommand{\subs}{\subseteq}
\newcommand{\wo}{\setminus}
\newcommand{\abs}[1]{\left|#1\right|}
\newcommand\set[2]{\left\{#1:#2\right\}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\N}{\mathbb{N}}

\begin{document}
\maketitle
\thispagestyle{empty}

\noindent
You will be graded on both the accuracy and the clarity of your solutions.
One purpose of the homework is to give you an opportunity to practice your proof-writing skills.

You are welcome\,---\,encouraged, even!\,---\,to collaborate on homework, but you should not 
copy solutions from any source, nor should you submit anything that you don't understand.

\begin{problem}
(First\,---\,not on your homework, just for yourself\,---\,write down the \emph{definition} of the 
absolute value function.)
 \begin{enumerate} [(a)]
  \item Prove the following formula for the maximum of two real numbers $x$ and $y$:
\[
 \max(x,y) = \tfrac12(x + y + \abs{x-y}).
\]
  \item Prove that the equality $\abs{ab} = \abs{a}\cdot\abs{b}$ holds for all real numbers $a$ and $b$.
  \item Prove, by first establishing the inequality $(a+b)^2 \le (\abs{a} + \abs{b})^2$ or otherwise,
  the \emph{triangle inequality}:\footnote{Meditate on why it is called this.}
  \[
   \abs{a+b} \le \abs{a} + \abs{b}.
  \]
  \item Prove that $\abs{a-c} \le \abs{a-b} + \abs{b-c}$ for all $a,b,c\in\R$. (\emph{Hint}: Use (c).)
 \end{enumerate}
\end{problem}

\begin{problem}		% stolen from 825
 Recall that you showed $\sqrt3$ to be irrational. The same proof shows that $\sqrt5$ is irrational.
 By considering their product or otherwise, prove that $\sqrt3 - \sqrt5$ and $\sqrt3 + \sqrt5$ are either both
 rational or both irrational. Deduce that they must both be irrational.
\end{problem}

\begin{problem}
For each item, compute the requested supremum or infimum or carefully explain why
it does not exist. Either way, \emph{prove} that your answer is correct.
\begin{enumerate} [(a)]
 \item Determine $\sup A$ for $A = \set{\tfrac{(-1)^n}{n} }{ n\in \mathbf{N}\wo\{0\} }$.
 \item Fix $\alpha\in (0,1)$. Determine $\inf(B)$ for $B = \set{ \alpha^n }{ n\in\N}$.
 \item Fix $\alpha\in (1,\infty)$. Determine $\sup(C)$ for $C = \set{ \alpha^n }{ n\in\N}$.
\end{enumerate} 
\end{problem}

\begin{problem}
Prove each of the following assertions.
\begin{enumerate} [(a)]
 \item If $x$ is an upper bound for a set $S\subseteq\mathbf{R}$ and $x$ is also a member of $S$,
 then $x = \sup S$.
 \item Suppose that $x$ is an upper bound for a set $S\subseteq\mathbf{R}$.
 Then $x = \sup S$ if and only if for every $n\in\mathbf{N}$ there is an element of $S$ in the interval
 $(x - \tfrac1n,x]$.
 
 \item If $A\subs B$ are nonempty, bounded sets of real numbers, then
 \[
  \inf B \le \inf A \le \sup A \le \sup B.
 \]
\end{enumerate}
\end{problem}

\begin{problem}
\begin{enumerate} [(a)]
\item
Suppose that $A,B\subs\R$ are nonempty and bounded above. Find a formula for $\sup(A\cup B)$
and prove that it is correct.
\item 
 Suppose that $\alpha>0$ and that $A\subs \R$ is nonempty and bounded above.
 Let $C = \set{ \alpha x}{ x\in A}$. Prove that $C$ is nonempty and bounded above and that 
 $\sup C = \alpha\sup A$.
\item
 Suppose that $A,B\subs\R$ are both nonempty and bounded above. Let 
 \[
  C = \set{ a + b }{ a\in A,\, b\in B}.
 \]
Prove that $C$ is nonempty and bounded above and that $\sup C = \sup A + \sup B$.
\end{enumerate}
\end{problem}

\begin{problem}
Give an example satisfying the requested condition or prove that no such example can exist.
 \begin{enumerate} [(a)]
  \item A sequence all of whose terms are (strictly) negative that converges to $0$.
  \item A sequence all of whose terms are (strictly) negative that converges to $0.01$.
  \item A sequence with infinitely many 0s that does not converge to 0.
  \item A sequence with infinitely many 0s that converges to a number $\ne0$.
  \item A sequence all of whose terms are non-integers that converges to an integer.
  \item A sequence all of whose terms are integers that converges to a number that is not an integer.
  \item Two non-convergent sequences $(a_n)$ and $(b_n)$ such that the sequence $(a_n+b_n)$ converges.
 \end{enumerate}
\end{problem}

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\end{document}




































