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{\bf Math 165: Honors Calculus I}
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{\bf Quiz 9}\quad {\sl Nov.\ 17, 1994}

\begin{enumerate}
\problemspace
\item
  \begin{enumerate}[a)]
    \item State the \textsc{Boundedness Theorem for Continuous Functions}.
    \answerspace

    \item  State the \textsc{Extreme-Value Theorem for Continuous Functions}.
    \answerspace
    
    \item Use a) to prove b). Hint: Let $M = \sup f$, $g(x) = M - f(x)$, and
    consider $1/g(x)$.
    \answerspace\answerspace

  \end{enumerate}


\problemspace
\item Let $f(x)$ be continuous on $[a,b]$ and suppose $0 < f(x) < 1$ for all 
  $x \in [a,b]$. Prove that there is a positive integer $n$ such that
  \[
    \frac{1}{n} \le f(x) \le 1 - \frac{1}{n}
   \]
   for all $x \in [a,b]$
  \answerspace\answerspace


\item Let $f(x)$ be continuous on $[a,b]$ and let $p \in [a,b]$.
Prove that for any $\varepsilon > 0$ there exists a subinterval of $[c,d]\subset
[a,b]$ with $p\in (c,d)$ such that the span of $f$ on $[c,d]$ is $< \varepsilon$.

\answerspace\answerspace


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