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\def\naturals{\Bbb N}

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   \par\medskip\noindent\thexprobno . \marginpar{(#1 points)} }
\newcommand{\doubleint}{\int\!\!\int }

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\begin{flushright}
  A. Miller\\
  M521 Final Exam\\
  May 19, 1994
\end{flushright}

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

\bigskip
%exam1

\xprob{25} Give the $\epsilon-\delta$ definition of continuity.
Prove directly from the definition that the function
$f:\reals\arrow\reals$ defined by $f(x)=x^2$ is continuous.

%\xprob{25} 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.)
%

%exam2

%\xprob{25} Suppose that $f:\reals^2\arrow\reals$ is $C^\prime$
%and $Df(x,y)\not=0$ for every $(x,y)$.  Prove that
%$f$ is one-to-one.

\xprob{25} Give an example of a sequence $f_n:\reals\arrow\reals$ of
differentiable functions such that $f_n$ converges uniformly to
a function $f$ which is not differentiable.  Be sure and prove that
the convergence is uniform.

% new stuff

\xprob{25} Suppose that $f:R\arrow\reals$ is continuous
on the rectangle $R$.  Prove the $f$ is Riemann integrable.

%\xprob{25} Suppose that $f_n:R\arrow \reals$ converges uniformly
%to a $f:R\arrow \reals$ and each $f_n$ is continuous on the rectangle
%$R$.  Prove
%$$\lim_{n\arrow\infty}\doubleint_Rf_n=\doubleint_R f.$$
%
\xprob{25} Prove the following.
Suppose $f:R\arrow \reals$ is continuous
on the rectangle $R=[a,b]\cross[c,d]$ and define $F:[c,d]\arrow\reals$
by $F(x)=\int^d_c f(x,y)dy$.  Then $F$ is Riemann integrable
and $$\int^b_aF(x)dx=\doubleint_R f.$$

\xprob{25} Prove the following.
Suppose $a_n>0$ for all $n$ and
$$\lim_{n\arrow\infty} {a_{n+1}\over a_n}=r.$$
Then $\sum a_n$ converges if $r<1$, and
$\sum a_n$ diverges if $r>1$.

\xprob{25} Prove that for any sequence $a_n>0$
$$\liminf_{n\arrow\infty}{a_{n+1}\over a_n}\leq
  \liminf_{n\arrow\infty}{^n\sqrt{a_n}}\leq
  \limsup_{n\arrow\infty}{^n\sqrt{a_n}}\leq
  \limsup_{n\arrow\infty}{a_{n+1}\over a_n}$$

\xprob{$0$-$\infty$} (Bonus Question)
State and prove any result from this class.
You will be graded on the basis of the difficulty of the result and the
clarity and correctness of your presentation.



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