= Solution
The <Dahlquist equivalence theorem> states that a <linear multistep method> has <convergence of a numerical method> \b[if and only if it is consistent and zero-stable], under the standard assumptions for the <initial value problem> and convergent starting values. <Consistency of a numerical method> means order at least one, or
$$
\rho(1)=0,\qquad \rho'(1)=\sigma(1),\quad \rho(\zeta)=\sum_{\ell=0}^s\rho_\ell\zeta^\ell,\quad \sigma(\zeta)=\sum_{\ell=0}^s\sigma_\ell\zeta^\ell.
$$
The <root condition for a multistep method> expresses <zero-stability>: every root of $\rho$ has modulus at most one, and every root with modulus one is simple. Usual hypotheses include a well-posed <ordinary differential equation> with $f$ continuous in time and <Lipschitz continuous> in the solution variable, a valid step equation, and starting approximations converging to the exact values.
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