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{\bf Supplementary problems (assigned 9/19/03)}
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\begin{enumerate}
\item 
Rudin's definition of a connected subset of a metric space is a little
non-standard.  Prove that his definition is equivalent to the following
more standard one: 

\emph{A subset $E$ of a metric space $X$ is connected if for every pair 
$U,V\subset X$ of disjoint non-empty open sets whose union contains $E$,
we have either $E\subset U$ or $E\subset V$. }

In other words show that $E$ satisfies Rudin's definition if and only if it
satisfies this one.
\item Let $P\subset [0,1]$ be the middle thirds Cantor set discussed
in Rudin and in class.  
\begin{enumerate}
\item
Show that $P$ is \emph{totally disconnected}.  That is, for
every two points $x\neq y$ in $P$, there are open sets $U,V\subset\R$ such
that $x\in U$, $y\in V$ and $P\subset U\cup V$.
\item
Compute the sum of the lengths of the open intervals in $[0,1] - P$.  
Based on your computation, if we were to assign a length
to the Cantor set itself, what would it have to be?
\item 
(Extra Credit) Show that the function 
$$
(a_j)_{j\in\N} \mapsto \sum_{j=1}^{\infty} \frac{2a_j}{3^j}.
$$
is a bijection from $2^\N$ onto $P$.  (Note that in some sense, this gives
a description of points $P$--a point $p\in[0,1]$ belongs to $P$ if and 
only if it is given as a ternary decimal expansion whose digits are all
zeroes and twos.)
\end{enumerate}

\item Show that the sequence $\{\sin n\}_{n\in\N}\subset \R$ diverges.
\end{enumerate}

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