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\def\naturals{\Bbb N}

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\newcommand{\xprob}[1]{\addtocounter{xprobno}{1}
   \par\medskip\noindent\thexprobno . \marginpar{(#1 points)} }

\begin{document}

\begin{flushright}
  A. Miller\\
  M521 Exam 1\\
  March 3, 1994
\end{flushright}

\bigskip

\begin{center}
  Keep this sheet.  Use your own paper for answers.\\
\end{center}

\bigskip

\begin{center}
  Results proved in class
\end{center}

\xprob{7} Prove that $\forall x,y\in \reals$ if $x<y$, then
$\exists r\in\rationals$ such that $x<r<y$.

\xprob{3} Define $p$ is a cluster point of the sequence
$\langle p_n:n\in\naturals\rangle$.

\xprob{3} Define $\langle p_n:n\in\naturals\rangle$ is a bounded sequence.

\xprob{7} Prove that every bounded sequence in $\reals$ has
a cluster point.

\xprob{3} Define $f$ is continuous at $p_0$ where
$f:D\arrow\reals^m$ and $p_0\in D\subseteq\reals^d$.

\xprob{3} Define $f$ is continuous where
$f:D\arrow\reals^m$ and $D\subseteq\reals^d$.

\xprob{3} Define $f(D)$ where $f$ is as above.

\xprob{3} Define $C\subseteq\reals^n$ is a compact set.

\xprob{7} Prove that if $f:D\arrow \reals^m$ is continuous and
$D\subseteq \reals^d$ is compact, then $f(D)$ is compact.

\bigskip

\begin{center}
  Exercises
\end{center}


\xprob{7} Prove that for every $p,q\in \reals^d$ that
$||p-q||\geq ||p|| - ||q||$.

\xprob{3} Define $\langle p_n:n\in\naturals\rangle$ is a Cauchy sequence.

\xprob{7} Give an example of a sequence $\langle x_n:n\in\naturals\rangle$
of real numbers such that
$$\forall \epsilon>0\;\; \exists N \in \naturals\;\;
\forall n\geq N \;\;|x_{n+1}-x_n|<\epsilon$$
but $\langle x_n:n\in\naturals\rangle$ is not a Cauchy sequence.
(Prove that your example is not Cauchy.)

\xprob{7} Prove that if $C\subseteq \reals$ is a nonempty closed
and bounded set,
then $C$ has a maximum element, i.e.,
$\exists x\in C\;\;\forall y\in C \;\;y\leq x$.


\bigskip

\begin{center}
  New problems
\end{center}

\xprob{3} Define $U\subseteq \reals^d$ is open in $\reals^d$.

\xprob{3} Define $C\subseteq \reals^d$ is closed in $\reals^d$.

\xprob{7} Prove (a) iff (b) where

(a) $C$ is closed in $\reals^d$

(b) $\forall p\in \reals^d\;\;$ if $\forall \epsilon>0 \;\;\exists q\in C$ with
$||p-q||<\epsilon$, then $p\in C$.

\xprob{7} Let $\rationals=\{r_n:n\in \naturals\}$ be any countable
enumeration of the rational numbers.  Define the function
$f:\reals\arrow\reals$ by the rule:
$$f(x)=\left\{\begin{array}{ll}
                0   & \mbox{if $x$ is irrational}\\
                1/n & \mbox{if $x=r_n$}          \\
              \end{array}\right. $$
Prove that $f$ is discontinuous at each rational and continuous
at each irrational.

\xprob{3} Define $f$ is uniformly continuous
where $f:D\arrow\reals^m$ and $D\subseteq \reals^d$.

\xprob{7} Suppose that $f:D\arrow \reals^m$ is uniformly
continuous where $D\subseteq\reals^d$ and $\langle p_n:n\in\naturals\rangle$
is a Cauchy sequence of elements of $D$.
Prove that
$$\langle f(p_n):n\in\naturals\rangle$$ is a Cauchy sequence.

\xprob{7} Suppose that $g:\reals\arrow \reals$ is a function and
$$G=\{(x,y)\in\reals^2: y=g(x)\}$$
is its graph.
Prove that if $g$ is continuous, then $G$ is closed.


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