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\textbf{Math 165: Honors Calculus I}

\textbf{Assignment 10}\quad \textsl{Sept. 22, 1999}

\begin{enumerate}
\problemspace \item Let $A$ and $B$ be subsets of real numbers each
of which has an infimum.  Let $C = \{ a + b \, | \, a\in A,\ b \in
B \}$. Prove that $C$ has an infimum and $\inf C = \inf A + \inf
B$. (The proof of the corresponding theorem for supremum,
Theorem~I.33~(a), is given on p.27.)

\problemspace
\item Define
  \[
  f(x) =\left\{\begin{array}{ll}
        0 &\mbox{if $x = 1/n$ for some $n\in \N$}\\
        1 &\mbox{otherwise}\\
        \end{array}\right.
  \]
Prove or disprove that $f(x)$ is
integrable on $[0,1]$.

\problemspace
\item Use Theorem 1.14 to find an approximation to the integral $\int_1^2
\frac{1}{x}\,dx$ so that the error is $< .05$. You may use a calculator or
computer, but you must prove the accuracy of your approximation.

\end{enumerate}

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