Theorem (Newton-Raphson Theorem).  Assume that and there exists a number , where .  If   , then there exists a such that the sequence defined by the iteration  

        for      

will converge to for any initial approximation  .  

Proof.

The geometric construction of    does not help in understanding why    needs to be close to  p  or why the continuity of    is essential.  
Our analysis starts the Taylor polynomial of degree n = 1 and its remainder term  

          

where  c  lies somewhere between    and  x.  Substituting  x = p  into this equation and using the fact that    produces  

        .  

If    is close enough to  p,  then the last term on the right side will be small compared to the sum of the first two terms.  Hence it can be neglected and we can use the approximation  

        .  

Solving for  p  in this equation, we get  .    This is used to define the next approximation    to the root  

        .  

When    is used in place of    in this equation, the general rule is established.  

        .  

    For most applications this is all that needs to be understood.  However, to fully comprehend what is happening, we need to consider the fixed-point iteration theorem, and apply it in our situation.  The Newton-Raphson formula is

        .  

Newton's method is merely fixed point iteration using the function  ,  starting with and using the iteration  

        .  

Let us recall the important theorem about fixed point iteration.

Theorem (Second Fixed Point Theorem). 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.  

The key in the proof for convergence in Newton's method is in the analysis of  :  

        .

By hypothesis,  ,  thus  .  
Because  ,  and    is continuous,  for any value ,  with  
it is possible to find a    so that the hypothesis  (i).  |,  is satisfied on  .  

Therefore, a sufficient condition for    to initialize a convergent sequence  ,  which converges to the root   of    is that    and that    be chosen so that  

          for all  .  

Q. E. D.

(c) John H. Mathews 2004