The method of successive approximations uses the equivalent integral equation for (1) and an iterative method for constructing approximations to the solution.  This is a traditional way to prove (1) and appears in most all differential equations textbooks.   It is attributed to the French mathematician Charles Emile Picard (1856-1941).  
    
Theorem 2 (Successive Approximations - Picard Iteration).  The solution to the I.V.P in (1) is found by constructing recursively a sequence    of functions
        ,   and
(2)    
        .
Then the solution to (1) is given by the limit:

(3)        .
Proof.

    Begin by reformulating (1) as an equivalent integral equation.  Integration of both sides of (1) yields

(4)        .  

Applying the Fundamental Theorem of Calculus to the left side of (4) yields    and we have  ,  which can be rearranged to obtain  

(5)        

    Observe that    occurs on both the left and right hand sides of equation (5).  We can use this formula and input    in the integrand   on the right and then output the next iteration for     on the left side.  This is a type of fixed point iteration, the most familiar form of which is Newton's method for root finding.  

    Start the iteration with the initial function     

        ,  
        
then define the next function    as follows  
        
        .  

Next   is used to construct    as follows  

        .  
        
The process is repeated, and once has been obtained, the next function is given recursively by  

(6)        .  

    We must take the limit as    in (6).  Assume that the limit (3) exists, then    and we write  

        

If there is "no problem" when taking limits on the right side then we might expect the following

            

This is the "intuitive proof"  of equation (5).                                  Q.E.D.

For More Proof.  More details for the existence and uniqueness of    can be found in textbooks and the literature.

1. Stephen W. Goode, Differential Equations and Linear Algebra, 3rd edition, Prentice Hall, Upper Saddle River, NJ, 2005, Appendix 4.
2. R. Kent Nagle; Edward B. Saff; and Arthur David Snider, Fundamentals of Differential Equations and Boundary Value Problems, 4th edition, Addison-Wesley, Boston, MA, 2004,  Chapter 13, Sections 1 and 2.
3. C. Henry Edwards; and David E. Penney, Differential Equations and Boundary Value Problems: Computing and Modeling, 3rd edition, Prentice Hall, Upper Saddle River, NJ, 2005, Appendix A.1.
4. William E. Boyce; and Richard C. DiPrima, Elementary Differential Equations and Boundary Value Problems, 7th Edition, John Wiley and Sons, New York, NY, 2002, Chapter 2, Section 8.
5. Garrett Birkhoff; and Gian-Carlo Rota,  Ordinary Differential Equations, 4rd Ed., John Wiley and Sons, New York, NY, 1989, Chapter 1, p. 23.   
6. Einar Hille,  Lectures on Ordinary Differential Equations, Addison-Wesley Pub. Co., Reading, MA, 1969, pp. 32-41.
7. Arthur Wouk, On the Cauchy-Picard Method, The American Mathematical Monthly, Vol. 70, No. 2. (Feb., 1963), pp. 158-162, Jstor.  

(c) John H. Mathews 2005