Theorem (Second Fixed Point Fheorem). Assume that the following hypothesis hold true.  
(a)     is a fixed point of a function ,
(b)    ,
(c)     is a positive constant,
(d)    , and
(e)      for all  .  
Then we have the following conclusions.
(i).    If   for all   ,  then the iteration    will converge to the
    unique fixed point .  In this case, is said to be an attractive fixed point.  
(ii).    If   for all   ,  then the iteration    will not converge to .  
    In this case, is said to be a repelling fixed point and the iteration exhibits local divergence.  

Proof.

We first show that the points all lie in .  Starting with , we apply the Mean Value Theorem.

There exists a value so that  

(1)          

        .  

Therefore,   is no further from than was, and it follows that  .  

In general, suppose that  ;  then  

(2)          

        .  
        
Therefore,   and hence, by induction, all the points    lie in .  

To complete the proof of (i), we will show that  

        .  

First, a proof by induction will establish the inequality  

        .  

The case follows from the details in relation (1), first step in our proof.  

Using the induction hypothesis and the ideas in (2), the second step in our proof, we obtain

        .  

Thus, by induction, inequality holds for all .  

Since , the term goes to zero as goes to infinity.  Hence  

        .  

The limit of is squeezed between zero on the left and zero on the right, so we can conclude that .  

Thus    and, the previous theorem showed that the iteration    converges to the fixed point .  

Therefore, statement (i) of the theorem is proved.  We leave statement (ii) for the reader to investigate.

Q. E. D.

(c) John H. Mathews 2004