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{\course}         \hfill {Russell Brown}\\ 
Test 
\medskip	


Answer 5 of the 6 questions below. Indicate clearly which of the
questions I should not grade.  Each question is weighted
equally.   

\begin{enumerate}

\item  If $E$ is a closed, bounded, nonempty set of real
numbers, prove that $ s= \sup E$ lies in $E$.



\item Consider the set of all sequences $ s=\{s_n\}$ for which 
take  only the values $0$ or $1$ and which eventually are zero.  That
is for each sequence $\{s_n\}$, there exists an $N$ so that if $n>N$, then
$s_n = 0$.  Is the set of all such sequences countable? 

\item Let $X$ be an infinite set and suppose the metric on $X$ is
given by
$$
d(p,q) = \left\{ \begin{array}{ll} 1, \qquad& p\neq q\\ 0, \qquad &
p=q.
		 \end{array}
\right. 
$$
Is the set $X$ connected?

\item  Suppose that $E$ is a compact set of real numbers, prove that
$E$ is bounded. You may not use the Heine Borel theorem.  Hint:
Consider the open sets $G_n = \{ s : |s| < n\}$.



\item If $\{a_n\}$  is a sequence of real numbers which satisfies
$
\lim _{n\rightarrow \infty} a_n = 0$ and  $ \{b_n\}$ is a bounded  sequence of
real numbers, prove that$ \lim _{n\rightarrow \infty} a_n b_n = 0$.


\item  \begin {enumerate} \item
 Complete the following sentence without using no, not or
never.  A sequence $\{ p_n\}$  in a metric space does not converge to
$p$ if \dots
\item Using  your answer from part a), show that the sequence $(-1)^n$
does not converge to 1.
       \end {enumerate}






\end{enumerate}

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