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\vspace{-1in}
\begin{center}
 \large\bf \m Math 365,  Test \# 2
 $\left.\hspace{2em}\oval(80, 40)\right/ 100$
 \end{center}
\begin{flushright}
Nov 6,  1996\\
Instructor: Bei Hu
\end{flushright}

\hspace{.5\textwidth} Name:\hrulefill


1. (25 points). True or False. (No proof or counter example needed).

(a). In a metric space, infinite subset of a bounded set has a limit point.
\vspace{1in}

(b). In a metric space, nonempty disjoint sets are separated.
\vspace{1in}

(c). In a metric space, a perfect set is uncountable.
\vspace{1in}

(d). In a compact metric space, a Cauchy sequence is convergent.
\vspace{1in}

(e). In a metric space, a Cauchy sequence is bounded.

\newpage
2. (20 points).  

(a). State PRECISELY the Heine-Borel Theorem.
\vspace{1.8in}

(b). State PRECISELY the definition of a connected set.
\vspace{1.8in}

(c). State PRECISELY the Monotone convergence Theorem for a
sequence $\{a_n\}$.
\vspace{1.8in}

(d). State PRECISELY the definition of the Cantor set.
\vspace{1.8in}
\newpage


3. (10 points). Let $E$ be a connected set. Prove that its
closure $\bar E$ is a connected set.
\vspace{1.6in}
\newpage


4. (10 points). Suppose that $0<a_1<2$ and $a_{n+1} = \sqrt{2{a_n}}$.
Prove that $\{a_n\}$ converges, and find the limit.


\newpage

5. (10 points)
  Let $X$ be a metric space and let $\{p_n\}$ be a Cauchy sequence.
Suppose that a subsequence of $\{p_n\}$ converges in $X$. Prove
that the sequence $\{p_n\}$ itself converges. 


\newpage

%6.
%(15 points). Let $X$ be a metric space. For fixed $\delta>0$ and $q\in X$,
%define $A=\{p| d(p,q)<\delta\}$, and $B=\{p| d(p,q)>\delta\}$. 
%
%i). Prove that $\bar A \subset \{p| d(p,q)\le \delta\}$. Is the
%equality valid? (One can similarly prove that
%$\bar B \subset \{p| d(p,q)\ge \delta\}$.)
%
%ii).  Prove that $A$ and $B$ are separated sets.
%
%iii). Suppose that $X$ is a connected metric space with more
%than two points $q_1$ and $q_2$. Prove that $X$ is uncountable.

6. (10 points)
 Let $X$ be a metric space, $\delta>0$ and $q\in X$.
 Define $A=\{p| d(p,q)<\delta\}$
 Prove that $\bar A \subset \{p| d(p,q)\le \delta\}$. Is the
 equality Vail? Prove or give a counter example.

\newpage

7. (5 points)
 Let $s_n= \dis \sqrt{n^2 + 3n} -n$. Find 
$$\lim_{n\rightarrow\infty} s_n$$ 


\newpage

8.  (10 points). Let $X$ be a connected metric space.
Suppose that $E$ is a nonempty subset of $X$ such that
$E$ is closed and open. Prove that $E=X$.

\newpage

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