define a solution of (8.11), and they also satisfy (8.13). If the method is convergent, (8.12) must hold. If , this immediately implies . If , we note that

and

Since the term on the left tends to zero as , the term on the right must do the same, which again implies . This proves the first part of the assertion of the theorem. In order to prove the second part, assume that is a root of of multiplicity exceeding 1. Then, again the numbers

(8.15)

represent a solution of (8.11). They also satisfy (8.13). Hence they must satisfy (8.12) for a convergent method. If or , we have for that , and it follows immediately that . If , we can make use of the relation

where . Since as in view of (8.12), the term on the left tends to zero as , and we conclude that . This proves the theorem.