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Homework for Math 103
Assignment One
Due Wednesday 6 October 1999


\begin{ques}
Let $(X,d)$\ be a metric space. The \emph{open ball of radius $r$ ...
...n and that the set $\tau$\ of open sets in $X$\ is a topology on $X$.
\end{ques}


\begin{ques}
Suppose that $f:[a,b]\to\mathbf{R}$\ is bounded and that both $\ma...
...al{Q}$, and then consider $\mathcal{R}:=\mathcal{P}\cup\mathcal{Q}$.)
\end{ques}


\begin{ques}
Prove that a bounded function $f:[a,b]\to\mathbf{R}$\ is Riemann i...
...quation*}
U(f,\mathcal{P})-L(f,\mathcal{P})<\epsilon
\end{equation*}\end{ques}

\begin{ques}
% latex2html id marker 33Using only the definition of Riemann in...
... make use of the theorem
that implies $f$\ is
uniformly continuous.
\end{ques}


\begin{ques}
(\emph{Rudin}: Page 31, \char93 5) Suppose that $f,g:(\mathcal{M})...
...Since $\infty-\infty$\ is undefined, $h$\ is not
everywhere defined.)
\end{ques}


\begin{ques}
% latex2html id marker 47Suppose that $Y$\ is either the extende...
..._{n}f_{n}(x)$\ exists is measurable. (Hint: use problem~\ref{prob8}.)
\end{ques}


\begin{ques}
Recall from calculus that if $\lbrace\,a_{n}\,\rbrace $\ is a sequ...
...y distributions
\lq\lq live on'' countable sample spaces.)
\end{enumerate}\end{ques}





Math 103 Fall 1999
1999-09-29