= Solution
Fix $0<r<1$ and $k\geq1$. On $|z|=1$ and $|z_0|<r$, the <reverse triangle inequality> gives $|z-z_0|\geq1-r$. Subtract the two <Cauchy integral formula> expressions and apply the <estimation lemma>, using circle length $2\pi$:
$$
\sup_{|z_0|<r}|f_n^{(k)}(z_0)-f^{(k)}(z_0)|\leq\frac{k!}{(1-r)^{k+1}}\sup_{|z|=1}|f_n(z)-f(z)|\longrightarrow0.
$$
Thus \b[every fixed-order derivative converges uniformly on each smaller disc]. The right-hand factor tends to zero by the assumed uniform convergence on $D$. Keeping a positive gap $1-r$ from the integration contour is the essential feature of this <Cauchy derivative convergence bound>. If $r\leq0$, the specified open disc is empty and the claim is vacuous.
Back to article page
