Theorem:

A stable and consistent linear multistep method is convergent.

Proof: Let the function satisfy the conditions of the existence and uniqueness, and let be an arbitrary constant. We shall denote by the solution of the initial value problem . Let be the solution of the difference equation (8.1), defined by the starting values

. We set

and assume that

(8.21)

We then have to show that for any

We begin by estimating the quantity , where denotes the difference operator defined earlier.

The function is continuous in the closed interval . We define for the quantity

For , we can write

where . Furthermore, since

where , we have