\section{(Explicit Methods)}
%We know the following for a convergent explicit linear multistep method
%\begin{enumerate}
%\item[a.] $\rho(1) = 0$
%\item[b.] $\rho(r) = 0 \Longrightarrow |r| \leq 1$
%\item[c.] $\rho(r) = 0$ and $|r| = 1 \Longrightarrow \rho'(r) \neq 0$
%\item[d.] $\sigma(1) = \rho'(1)$
%\end{enumerate}
The stability polynomial is defined as
\[\pi(r,\kappa) = \rho(r) - \kappa\sigma(r)\]
We have to prove that as $\kappa \rightarrow \infty$, we cannot have any of the roots of $\pi(r,\kappa) \rightarrow \infty$ where the roots $r$ depend on $\kappa$.  We know from the Schur criterion that for a region of stability we have
\[|\alpha_{0} - \kappa\beta_{0}| < 1\]
Which puts clear bounds on the region surrounding $\kappa$.  However, it is unclear what happens when $\beta_{0} = 0$.  To be honest, I don't know how to procede from this point.  Please don't penalize the other members of my group for my inability to solve this problem.  \em Jeremy Morris\em.

%If $\beta_{0} = 0$, we know that one of the roots of $\pi$ is $0$, and we can reduce the polynomial by one degree and apply the Schur criterion again to verify that all roots are less than one in absolute value.  All we need to show now is that all $\beta_{j} \neq 0$.  We can see this by looking at the generalized explicit linear multistep methods
%\[\sum_{j=0}^{k}\alpha_{j}y_{n+j} = h\sum_{j=0}^{k-1}\beta_{j}f_{n+j}\]
