\news[Supplementary exercises]

Here are some supplementary exercises.
 \newcounter{exnumber}
\setcounter{exnumber}{\value{exercise}}
\let\mytheexercise\theexercise
\renewcommand{\theexercise}{\Alph{exercise}}
\setcounter{exercise}{0}

\begin{exercise}
  Prove that \tech{definition by recursion} is valid.  That is, if $S$
  is a set, and $b\in S$, and $f:\N\times S\to S$ is a function, then
  there is a \emph{unique} sequence $(a_n:n\in\N)$ such that:
  \begin{itemize}
    \item
$a_0=b$, and
\item
$a_{n+1}=f(n,a_n)$ for all $n$ in $\N$.
  \end{itemize}
\end{exercise}

\begin{exercise}
  By recursion, define the powers $x^n$, where $n\in\N$ and $x$ is in
  a field. 
\end{exercise}

\begin{exercise}
  Define $n!$ by recursion.
\end{exercise}

\begin{exercise}
Prove that $x^{n+1}-y^{n+1}=(x-y)\sum_{k=0}^nx^{n-k}y^k$ for all $n$
in $\N$ and all $x$ and $y$ in a field.
\end{exercise}

\begin{exercise}
Prove that the set $\R\setminus\Q$ of \defn{irrational numbers} is
dense in $\R$.  
\end{exercise}

\begin{exercise}
  Prove that if $x<0$, then there is \emph{no} real number $y$ such
  that $y^2=x$.
\end{exercise}

\begin{exercise}
  Prove that for every \emph{positive} real number $x$ there is a
  \emph{unique} positive real 
  number $y$ such that $y^2=x$.  (The number $y$ is then denoted
  $\radix x$.)
\end{exercise}

\begin{exercise}
  If $n$ is a positive integer, and $\radix n$ is not an integer, then
  $\radix n$ is irrational.
\end{exercise}

\begin{exercise}
  Let $(F_n:n\in\N)$ be the \defn{Fibonacci} sequence, defined recursively
  by:
  \begin{itemize}
    \item
$F_0=F_1=1$, and
\item
$F_{n+2}=F_n+F_{n+1}$.
  \end{itemize}
Let $\phi_0$ and $\phi_1$ be the roots of the polynomial $X^2-X-1=0$.
Show that $\phi_1=-1/\phi_0$, and show that 
$$F_n=\frac{\phi_0^{n+1}-\phi_1^{n+1}}{\phi_0-\phi_1}.$$
\end{exercise}

\begin{exercise}
  Find the extrema of the following subsets of $\R$:
  \begin{itemize}
    \item
$\{11^a+13^b:a,b\in\Z\land a,b<0\}$;
\item
$\{x:x^2-x+1<0\}$;
\item
$\{x:(x-17)(x-3)(x+1)(x-19)<0\}$.
  \end{itemize}
\end{exercise}

\begin{exercise}
  If $A$ and $B$ are subsets of $\R$ bounded above, and are bounded
  below by $0$, prove that
  $\{ab:a\in A\land b\in B\}$ is bounded above, with supremum $\sup
  A\sup B$.
\end{exercise}

\let\theexercise\mytheexercise
\setcounter{exercise}{\value{exnumber}}