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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.

I'm having trouble how to proceed further.


marked as duplicate by Martin R, Kavi Rama Murthy, Shailesh, KReiser, Leucippus Dec 31 '18 at 2:04

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