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\pagestyle{fancy} \lhead{{\sf Homework 4 $|$ Due May 29 (Tuesday) }}
\rhead{\thepage} \cfoot{{\sf MthSc 453 $|$ Real Analysis I $|$ Summer
    I 2012 $|$ M.~Macauley}}


\begin{document}

$\;$ 

\noindent\emph{Read}: Rudin, Chapter 2, pages 24--30. \\

\begin{enumerate}

\item Prove that the Principle of Induction implies the Well Ordering
  Princple for $\N$. \emph{Hint}: Let $L(n)$ be the statement ``if
  $A\subset\N$ that contains a number $m\leq n$, then $A$ has a least
  element,'' and induct on $n$. \\



\item A complex number $n$ is said to be \emph{algebraic} if there are
  integers $a_0,\dots,a_n$, not all zero, such that
  \[
  a_nz^n+a_1z^{n-1}+\cdots+a_1z+a_0=0\,.
  \]
  \begin{enumerate}
  \item Prove that the set $\A$ of algebraic numbers is
    countable. \emph{Hint}: For every positive integer $N$, there are
    only finitely many equations with
    \[
    n+|a_0|+|a_1|+\cdots+|a_n|=N\,.
    \]
  \item Prove that there exist real numbers which are not algebraic. \\
  \end{enumerate}
  
   

\item Is the set of all irrational real numbers countable? Prove or
  disprove. \\


\item For $x,y\in \R$, define
  \[
  d_1(x,y)=(x-y)^2,\quad\; d_2(x,y)=|x^2-y^2|,\quad\;
  d_3(x,y)=|x-2y|,\quad\; d_4(x,y)=\frac{|x-y|}{1+|x-y|}.
  \]
  Determine, for each of these, whether it is a metric or not. That
  is, either verify that the three required conditions hold, or show
  by example that one of them fails. \\



\item Construct a bounded set of real numbers with exactly three limit
  points.
    
\end{enumerate}

\end{document}
