Module 8: Linear Multistep Methods
  Lecture 29: Necessary & Sufficient Conditions for Convergence
 

 

where . Hence it follows that

where . Since is assumed to be consistent, we have

Hence

(8.22)

where

We now subtract from the corresponding relation

satisfied by the values . Writing and setting

we get

where . In view of the Lipschitz condition,

The following Lemma is now used in the remaining part of the proof of the theorem.