How can I prove with using estimates of Cauchy: a sequence of holomorphic functions {$f_n$} uniformly converges to the holomorphic function f on D $\subset$ $\mathbb{C}$. Then for each compact set K $\subset$ D sequence {$f_n^{(k)}$} uniformly converges D to function $f^{(k)}$.

I prove estimates of Cauchy ($|f^{(n)}(z)|$ $\leq$ $\frac{n!M}{r^n}$). And I know that compact set means that set is closed and limited.

Thank you.

  • 1
    $\begingroup$ Hint : Proove that $f_n^{(k)}$ is cauchy-uniform. $\endgroup$ – C. Dubussy May 4 '16 at 10:34
  • $\begingroup$ if $f_n$ converges uniformly to $f$ on an open $U$, then $\int_\gamma f(z) = 0$ for every (finite length) contour $\gamma \subset U$, hence $f$ is holomorphic on $U$. is there another way for proving it ? $\endgroup$ – reuns May 4 '16 at 10:50

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