8. Proof of theorem 1.

We need the fundamental lemma of calculus of variations:

Lemma: Let f be a function in C[a,b] and assume that

for every


Proof of theorem 1: Let y be an extremal and let

Then  is an admissible and competing function for  Consider the functional 

Since y  is an extremal we must have

If we move the derivative inside the integral we get

This means that

for all

We integrate the second term by parts and get



this implies that

We thus have

for all

If we finally use the lemma above we can conclude that

The proof is complete.