Theorem 2: A necessary condition for convergence of the linear multistep method defined by (8.11) is that the order of the associated difference operator be at least 1.

The condition that the order is called the condition of consistency. In terms of the constants introduced earlier (in associated difference operator) the condition is equivalent to ; in terms of the polynomials , the condition of consistency is expressed by the relations

(8.16)

Proof: We begin by showing that . If the method is convergent, it is convergent in the initial value problem , with the exact solution . The difference equation (8.1) again reduces to

(8.17)

Assuming that the method is convergent, the solution of (8.17) assuming the exact starting values must satisfy as . Since in this case does not depend on , this is the same as saying that as . Letting in (8.17), we obtain . This is equivalent to . It follows that .