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
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
We integrate the second term by parts and get
this implies that
We thus have
If we finally use the lemma above we can conclude that
The proof is complete.