Cauchy Theorem for a triangle

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Theorem[edit]

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

Proof[edit]

Assume

.

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

and

(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

implies

which can be rewritten as

implies

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

Giving

Since can be chosen arbitrary small, then .