Theorem (First Fixed Point Theorem). Assume that , i. e.    is continuous on  .  
Then we have the following conclusions.
(i).    If the range of the mapping satisfies for all , then   has a fixed point in .
(ii).    Furthermore, suppose that is defined over and that a positive constant exists with
      for all  ,  then has a unique fixed point in .    

Proof of (ii).

Now we must show that the solution in (i) is unique.  

By way of contradiction, let us make the additional assumption that there exists  two fixed points   and  .

Now apply the Mean Value Theorem, and conclude that there exists a number    so that

    

Next, use the facts that    and    to simplify this expression and get

    .

But this contradicts the hypothesis in (ii) that   for all  .  

Thus, it is not possible for fixed points to exist.  Therefore, has a unique fixed point in .  

Q. E. D.

(c) John H. Mathews 2004