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We begin by stating a lemma:
Lemma: Let be an analytic function on some closed disk which has center and radius . Let denote the the boundary of the disk. It holds that
Proof: By the Cauchy integral formula we have that
The equation of a circle of radius and centre is given by . Hence,
and the proof is complete.
Something quickie: Given the assumptions in Gauss’ MVT, we have
The proof of the result is pretty straight forward by using the fact that
Back to the problem the result now follows by the lemma.
Tags: Integral, Real Analysis
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