Let be a domain, a differentiable function. Let be a triangle such that . Then
It will be shown that .
First, subdivide into four triangles, marked , , , by joining the midpoints on the sides. Then it is true that
Choose such that
Defining as , then
(where describes length of curve).
Repeat this process of subdivision to get a sequence of triangles
Claim: The nested sequence contains a point . On each step choose a point . Then it is easy to show that is a Cauchy sequence. Then converges to a point since each of the s are closed, hence, proving the claim.
We can generate another estimate of using the fact that is differentiable. Since is differentiable at , for a given there exists such that
which can be rewritten as
For we have , and so, by the Estimation Lemma we have that
As is of the form it has an antiderivative in D, and so , and the above is then just
Since can be chosen arbitrary small, then .