\section{Dense Orderings}

\note[Note]{This section is fairly difficult. If you get stuck here, try the bonus problems on the next page.}

\definition{}
An \textit{ordered set} is a set with an \say{order} attached to it. \par
A few examples are below:
\begin{itemize}
	\item $\mathbb{Z}$ is an ordered set under $<$.
	\item $\{\texttt{A}, \texttt{B}, ..., \texttt{Z}\}$ is an ordered set under $\diamond$,\par
	Where $\alpha \diamond \beta$ holds iff the letter $\alpha$ comes before letter $\beta$ in the alphabet.
\end{itemize}


\definition{}
We say an ordered set $A$ is \textit{dense} if for any $a, b \in A$ there is a $c \in A$ so that $a < c < b$.\par
Intuitively, this means that there is an element of $A$ between any two elements of $A$.

\problem{}
Show that the ordered set $(\mathbb{Q}, <)$ is dense.
\vfill

\problem{}
Show that the ordered set $(\mathbb{R}, <)$ is dense.
\vfill

\problem{}
Show that there is a real number between every two rationals.
\vfill

\problem{}
Show that there is a rational number between every two reals.
\vfill

\pagebreak