Cauchy derivative convergence bound (${\sup }_{|z|<r}|f_n^{(k)}-f^{(k)}|\le \frac{k!}{(R-r{)}^{k+1}}R{\sup }_{|z|=R}|f_n-f|$)

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.