This is similar to the maximum-modulos theorem. By Cauchy's theorem we know that:

Therefore, (by definition of contour integral)

Thus,

We have shown that on maximum value of on the disk is at least . However, and so the maximum value of must be equal to . Therefore, it follows that,

The only way that integral can equal to is it at each on we have .

Thus, for .

But was arbitrary and by shrinking and expanding we get the whole disk with .