\documentclass[11pt]{amsart}
%\usepackage{amsmath}

\newcommand{\R}{{\mathbb R}}
\newcommand{\N}{{\mathbb N}}

\begin{document}

{\large Math 131 --- Homework 3 }

\bigskip
\bigskip
%PART ALPHA.

\bigskip
\bigskip

{\bf READ Problem D.}
Let $u$ be an upper bound of non-empty set $A$ in $\R$.
Prove that $u$ is the supremum of $A$
if and only if for all $\epsilon > 0$ there is an $a \in A$ such that
$u-\epsilon < a$.

\medskip
%(We discussed this fact in class but now I am asking you to prove it!

Note that to show that ``$S$ if and only if $T$'' you must show
that $S$ implies $T$, and $T$ implies $S$.)

\bigskip
\bigskip


{\bf READ Problem E.} 
Let $A, B$ be nonempty subsets of $\R$ that are bounded above, 
and let $A+B=\{ a+b : a\in A, b\in
B\}$.  Show that
$$
\sup (A+B) = \sup A + \sup B.
$$

 
\bigskip
\bigskip


{\bf Problem F.} 
Let $A, B$ be nonempty subsets {\em of positive real numbers} that are bounded above, 
and let $A\cdot B=\{ a b : a\in A, b\in B\}$.  
Show that
$$
\sup (A\cdot B) = \sup A \sup B.
$$

\bigskip
\bigskip

{\bf Problem G.} 
(a) Let $A$ be a nonempty subset of $\R$ and suppose that $s = \sup A$ belongs to
$A$.  If $b$ is not in $A$, show that $\sup (A \cup \{ b\} )$ is equal
to the larger of the two numbers $s$ and $b$.

\medskip

(b) Use this to show that a nonempty finite set $A$ contains its
supremum.  [Hint--- use induction: show it is true first for a
  one-element set, then show that {\em if} it is true for an $n$-element set
  then it must be true for an $(n+1)$-element set.]

\bigskip
\noindent
%Do also {\bf Chapter 1 ( 12 )}.


%PART BETA.


\bigskip
\noindent
Do also {\bf Chapter 1 ( 6ab, 6cd, 12, R13, 15 )}.

\bigskip
\bigskip
\bigskip

\bigskip
Comment:  When you are asked a question, e.g., problem 1.15, you
should always give justification.

\end{document}
