= Cauchy derivative convergence bound
{c}
{title2=$\sup_{|z|<r}|f_n^{(k)}-f^{(k)}|\leq\frac{k!}{(R-r)^{k+1}}R\sup_{|z|=R}|f_n-f|$}
For $0<r<R$, suppose $f_n,f$ are holomorphic on a neighborhood of the closed radius-$R$ disc. Subtract their <Cauchy integral formula> expressions on that circle. Its length is $2\pi R$, and the distance from an interior point with modulus at most $r$ to the circle is at least $R-r$. The <estimation lemma> gives the displayed bound. Uniform convergence on that circle therefore implies uniform convergence of each fixed-order derivative on the smaller disc. The positive separation from the contour is essential for a uniform bound.
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