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\begin{document}

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\lhead{\bf HOMEWORK 6}
\chead{\bf  DUE:  Mon., May 7}

\rhead{\bf  NAME: \hspace{80} }
\noindent

\cfoot{\thepage}
 {\bf DIRECTIONS:}  

\begin{itemize}                 
         \item Turn in your homework as {\bf SINGLE-SIDED} typed or handwritten pages.  
         \item {\bf STAPLE} your homework together.  Do not use paper clips, folds, etc.
	\item {\bf STAPLE} this page to the front of your homework.
         \item {Be sure to write your name on your homework.}
	\item Show all work, {\bf clearly and in order}.
\end{itemize}

{\bf You will lose point 0.5 points for each instruction not followed.}

\bigskip\bigskip
\centerline{\begin{tabular}{|c||c|c|}\hline
& & \\
Questions & \quad Points \quad & \qquad Score \qquad\,\\ \hline\hline
& & \\
1 & 2 & \\ \hline
& & \\
2& 2 & \\ \hline
& & \\
3& 1 & \\ \hline
& & \\
4& 2 & \\ \hline
& & \\
5& 2 & \\ \hline
& & \\
6& 1 & \\ \hline
\hline
& & \\
Total& 10 & \\
\hline
\end{tabular}}

\newpage

{\bf Problem 1:}  (2 points) Let $x \in \mathbb{R}$, Show that there exists a sequence $s$ such that $s_k \in \mathbb{Q}$ for all $k \in \mathbb{N}$ and $\lim_{n \to \infty} s_k = x$.

{\bf Problem 2:} (2 points) Show that for any $a$, $b \in \mathbb{Q}$, we have $||a|-|b|| \le |a-b|$.

{\bf Problem 3:} (1 point) Show that the sum of two Cauchy sequence in $\mathbb{Q}$ is a Cauchy sequence in $\mathbb{Q}$.

{\bf Problem 4:}  (2 points) Show that addition is well defined in $\mathbb{R}$.

{\bf Problem 5:}  (2 points) Let $\left( a_k \right)_{k \in \mathbb{N}}$ be a Cauchy sequence of rational numbers such that $\left( a_k \right)_{k \in \mathbb{N}} \notin \mathcal{I}$.  Define the inverse sequence, $\left( b_{k} \right)_{k \in \mathbb{N}}$, by 
$$
b_k = \Big\{
\begin{array}{ll}
1, & \textrm{for} \ \ k \le N, \\
1/a_k, & \textrm{for} \ \ k > N,
\end{array}
$$
where for $n > N$ we know there exists an $r \in \mathbb{Q}^+$ such that $|a_k| > r$.  Prove that $\left( b_k \right)_{k \in \mathbb{N}}$ is Cauchy.

%{\bf Problem 6:} (0.5 points) Let $\left( a_n\right)_{n \in \mathbb{N}}$ and $\left( b_n\right)_{n \in \mathbb{N}}$ be Cauchy sequences of real numbers.  Prove that $\left(a_n \cdot b_n \right)_{n \in \mathbb{N}}$ is also Cauchy.

{\bf Problem 6:}  (1 point) Prove that every bounded sequence in $\mathbb{R}$ has a convergent subsequence.


\end{document}