In order to show that , consider the initial value problem . The exact solution is . The difference equation (8.1) now reads

(8.18)

For a convergent method every solution of (8.18) satisfying

(8.19)

where , must also satisfy

(8.20)

For a convergent method we may further more assume that

in view of the previous theorem. Let the sequence be defined by , where

This sequence obviously satisfies (8.19) and is easily shown to be a solution of (8.18).

From

,

we conclude that . This is equivalent to . This completes the proof of the theorem.