Suppose that g is continuously differentiable on some interval (c, d) that contains the fixed point p of g. Show that if , then there exists such that if , then the fixed-point iteration converges.
If , then by definition of the derivative, for a sufficiently small . Since , this means . So define such that , and we have just shown that , which is exactly the definition of convergent. So when , the value of each iteration converges to .
Notice that this proof allows you to predict the value of for a given function. Notably, for some fixed point where , if the approximation of at some point , then and the iteration of any point in the interval will converge on .