# Uniformly bounded sequence of holomorphic functions on $D$ such that for $a\in D$, $\lim_{n\rightarrow\infty}f_n^{(k)}(a)=0$ [duplicate]

Let $$f_n$$ be a uniformly bounded sequence of holomorphic functions on $$D$$. Suppose there exists a point $$a\in D$$, such that $$\lim_{n\rightarrow\infty}f_n^{(k)}(a)=0$$ for each $$k$$. Show that $$f_n\rightarrow0$$ uniformly on each compact subset of $$D$$.
Since $$f_n$$ is holomorphic and bounded, by Cauchy estimate, $$\displaystyle f_n^{(k)}(a)\leq \dfrac{k!\sup_{z\in D}|f|}{r^k}$$ where $$r$$ is the radius of the region $$D$$. Hence, $$f_n^{(k)}$$ is also bounded.