Proof

Theorem 3 (Fixed Point Theorem) says the sequence converges to . The Mean Value Theorem then means that

> expr:=p[n+1]-p;

> expr=g(p[n])-g(p);

> expr=D(g)(eta[n])*(p[n]-p);

Since lies between and , this sequence must also converge to so that . Hence,

> limit((p[n+1]-p)/(p[n]-p),n=infinity)=limit(D(g)(zeta[n]),n=infinity);

which equals . And therefore also,

> limit(abs(p[n+1]-p)/abs(p[n]-p),n=infinity)=abs(D(g)(p));

so that Fixed Point Iteration exhibits linear convergence when .