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\pagestyle{fancy} \lhead{{\sf Homework 2 $|$ Due May 22 (Tuesday) }}
\rhead{\thepage} \cfoot{{\sf MthSc 453 $|$ Real Analysis I $|$ Summer
    I 2012 $|$ M.~Macauley}}


\begin{document}

$\;$

\noindent\emph{Read}: Rudin, Chapter 1. Do not wory about understanding every
detail, but get a good sense of the big ideas. \\


\begin{enumerate}


\item Prove Proposition 1.15 in Rudin: The axioms for multiplication
  (see p.~5) imply the following statements.
  \begin{enumerate}
    \item If $x\neq0$ and $xy=xz$ then $y=z$. 
    \item If $x\neq 0$ and $xy=x$ then $y=1$.
    \item If $x\neq 0$ and $xy=1$ then $y=1/x$.
    \item If $x\neq 0$ then $1/(1/x)=x$. \\
  \end{enumerate}



\item Let $E$ be a nonempty subset of an ordered set. Suppose $\alpha$
  is a lower bound of $E$ and $\beta$ is an upper bound of $E$. Prove
  that $\alpha\leq\beta$. \\



\item Let $A$ be a nonempty set of real numbers which is bounded
  below. Let $-A$ be the set of all numbers $-x$, where $x\in
  A$. Prove that $\inf A=-\sup(-A)$. \\



\item For any real number $a\in\R$ and nonempty $B\subset\R$, define
  the set $a+B=\{a+b:b\in B\}$. Show that if $B$ is bounded
  above, then $\sup(a+B)=a+\sup B$. \\



\item Let $u$ be an upper bound of a 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 $u-\epsilon<a$. Formulate (but do not
  prove) an analogous statement for the infimum of $A$. \\



\item Let $A$, $B$ be subsets of $\R$ that are bounded above, and let
  $A+B=\{a+b:a\in A,\,b\in B\}$. Show that
  \[
  \sup(A+B)=\sup A+\sup B.
  \]


\end{enumerate}

\end{document}
