Hence the norm of the last two expressions on the R.H.S. of (5.7) can be bounded by . Substituting these in (5.7), we get

(5.8)

This is a difference equation of the type given in Lemma in Module 2 from which we have

This converges to zero as and , so the numerical solution converges to the solution of (5.5). Sufficiency of the condition follows immediately.

If, on the other hand, we have convergence, then the solution of (5.5), is identical to , the solution of .

Suppose also that and differ at some point . If we consider the initial value problem starting from , we have

leading to a contradiction. Hence the theorem.

Application of the above theorem: