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\begin{Large}

{\bf MATH 335: Real Analysis I -- FALL 2000}\\

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\begin{Large}

\centerline{\bf EXAM I}

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\hspace{2cm}



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\medskip

This  Exam contains 6 problems on 8 sheets of paper, including

this front cover.  You have to solve problems 1, 2 and three out

of four remaining problems. Do all of your work on the paper

provided.

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\begin{tabular}{|c|c|c|}

\hline \hline Problem & Possible points & SCORE \\ \hline \hline

1 & 30 & \\ \hline 2 & 25 & \\ \hline 3 & 15 & \\ \hline 4 & 15 &

\\ \hline 5 & 15 & \\ \hline 6 & 15 & \\ \hline \hline

TOTAL & 100 & \\

\hline \hline

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\noindent{\bf Problem 1.}

\medskip

\noi Do three of the following five (10 points each).



\medskip



\noi (a) State the definition of the Cauchy sequence.

\vspace{4cm}



\noi (b) State the definition of the divergent sequence.

\vspace{4cm}



\noi (c) Give the definition of the intersection of two sets.

\vspace{4cm}



\noi (d) State your favorite theorem about limits of sequences.

\vspace{4cm}



\noi (e) Give the definition of a countable set.

\vspace{4cm}



\newpage



\noindent{\bf Problem 2.}

\medskip

\noi Do five  of the following seven (5 points each). Explain your

answers. If necessary, provide a counterexample.



\medskip



\noi (a) True or false: if $\{a_n\}$ and $\{b_n\}$ are divergent

sequences, then $\{a_n b_n\}$ diverges.

 \vspace{5cm}



\noi (b) Suppose for every natural $n$

\[ \frac{n-2}{n+1} \le a_n \le \frac{n^3 + n^2}{n^3 + 7}\ .

\]

Find $\lim_{n\to\infty} a_n$ .

 \vspace{5cm}



\noi (c) True or false: if $\{a_n\}$ converges, then

$\sup\{a_n\}=\lim_{n\to\infty} a_n$ \vspace{5cm}



\noi (d) What is the domain of $f+g$, if $f$ and $g$ are functions

with domains $Dom(f)$, $Dom(g)$ resp. \vspace{5cm}



\noi (e) True or false: every bounded sequence is convergent.

\vspace{5cm}



\noi (f) True or false: a subsequence of a Cauchy sequence is a

Cauchy sequence. \vspace{5cm}



\noi (g) True or false: if $a$ is a greatest upper bound of the

set $S$ then $a$ belongs to $S$. \vspace{5cm}



\newpage









\noindent

{\bf Problem 3.} (15 points)

\medskip

\noi

Prove that for any natural $n$

\[

1^2 + 2^2 + \cdots + n^2 = \frac{n(n+1)(2n+1)}{6}\ .

\]



\newpage





\noindent

{\bf Problem 4.} (15 points)

\medskip



\noi Give an example of a {\bf divergent} sequence that has {\bf

exactly} one limit point.



\newpage





\noindent

{\bf Problem 5.} (15 points)



\noi Let $\{x_n\}$ be defined by

\[

x_1=2,\  x_{n+1} =\frac{3}{4-x_n}\  \mbox{for} \ n\ge 1\ .

\]

Prove that $\{x_n\}$ has a limit and find it.

\newpage



\noindent {\bf Problem 6.} (15 points)

\medskip



\noi Show using the definition of the limit that

\[

\lim_{n\to\infty} 2^{-\sqrt n} =0\ .

\]







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