 Solution

The Dahlquist equivalence theorem states that a linear multistep method has convergence of a numerical method 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, orThe 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.