Cauchy Theorem for a triangle

Theorem edit

Let   be a domain,   a differentiable function. Let   be a triangle such that  . Then


Proof edit



It will be shown that  .

First, subdivide   into four triangles, marked  ,  ,  ,   by joining the midpoints on the sides. Then it is true that


Giving that


Choose   such that


Defining   as  , then


(where   describes length of curve).

Repeat this process of subdivision to get a sequence of triangles


satisfying that

  and  .

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


Notice that




Since   can be chosen arbitrary small, then  .