Proof Sketch
By using the Cauchy integral theorem, one can show that the integral over C (or the closed rectifiable curve) is equal to the same integral taken over an arbitrarily small circle around a. Since f(z) is continuous, we can choose a circle small enough on which f(z) is arbitrarily close to f(a). On the other hand, the integral
over any circle C centered at a. This can be calculated directly via a parametrization (integration by substitution) where 0 ≤ t ≤ 2π and ε is the radius of the circle.
Letting ε → 0 gives the desired estimate
Read more about this topic: Cauchy's Integral Formula
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