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{\bf Math 165: Honors Calculus I}
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{\bf Quiz 1}\quad {\sl Sep. 3, 1998}

\begin{enumerate}

\item
  \begin{enumerate}[a)]
  \item Prove using only the axioms for the real numbers that
        $a\cdot 0 = 0$ for any $a\in\R$.
  \answerspace
  \item Define $-c$ for a real number $c\in \R$.
  \answerspace
  \item Prove using only the axioms for the real numbers  and parts a) \&
   b) that
        $(-a)b = -(ab)$ for any real numbers $a$, $b\in\R$.
  \answerspace
  \end{enumerate}

\problemspace
\item Give precise definitions for the following:

  \begin{enumerate}[a)]
  \item an inductive set, $S$.
  \answerspace

  \item the positive integers, $\N$.
  \answerspace

  \item $\sum_{k=1}^n a_k$.
  \answerspace
  \end{enumerate}

\problemspace
\item Use the Principle of Mathematical Induction to prove
\[
\sum_{k=1}^n k^3 = \Bigl(\sum_{k=1}^n k\Bigr)^2
\]
\answerspace\answerspace

\end{enumerate}

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