% LaTeX file -- AM/PM 331, Fall 2011, Assignment 1

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\begin{center}
\textbf{\large AMath/PMath 331 \\ Assignment 2\\
Due Monday October 3}
\end{center}
\vskip 2em 

\begin{Ex}

\item \begin{parts}
\item For each of the following sets, provide a sketch. State whether it is open, closed or neither.
If it is \textit{not closed}, identify a limit point of the set which is not in the set.
If it is \textit{not open}, identify a point in the set which is a limit point of the complement.
\vspace{.5ex}
\begin{itemize}
\item[(i)] $A = \{ (e^{-x}\cos x, e^{-x} \sin x) : x \ge 0 \} \cup \{(x,0) : 0 \le x < 1 \}$.
\item[(ii)] $B = \{ (r,s) : r,s \in \bQ,\ r^2+s^{-2}  < 1 \}$.
\item[(iii)] $C = \{ (x, \sin y) : x^2 < 4,\ y > 0 \}$.
\item[(iv)] $D = \{ (x,y) : x \ne 0,\  |y| < x^{-1} \}$.
\end{itemize}
\item Find the closure of each set, and decide whether this closure is compact.
\end{parts}

\item
\begin{parts}
\item Prove that if $U$ and $V$ are open subsets of $\bR^n$, then $W = U\cap V$ is also open.
\item Prove that if $\{ U_i : i \in I \}$ is a (possibly infinite) collection of open sets,
then the union $U = \bigcup_{i\in I} U_i$ is open.
\end{parts}

\item If $(\Bx_k)$ is a sequence in $\bR^n$ and $\dlim_{k\to\infty} \Bx_k = \Bx$,
prove that $\dlim_{k\to\infty} \|\Bx_k\| = \|\Bx\|$.

%\item Show that a subset $A$ of $\bR^n$ is complete if and only if it is closed.

\item  Suppose that $(\Bx_k)$ is a sequence in $\bR^n$ and 
$\dsum_{k=1}^\infty \| x_{k+1} - x_k \| < \infty$. \\
Prove that $(\Bx_k)$ is a Cauchy sequence.\\
\hint if $\ep>0$, there is an $N$ so that $\dsum_{k=N}^\infty \| x_{k+1} - x_k \| < \ep$.

\item Show that if $A \subset \bR^m$ and $B \subset \bR^n$ are compact sets,
then 
\[ A\times B = \{(a,b) \in \bR^{m+n} : a \in A \AND b \in B \} \]
is compact. 
\hint take a sequence in $A \times B$, find a subsequence based on the 
first coordinate, and then find a sub-subsequence based on the second coordinate. 


\end{Ex}
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