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Let $h_{n}$ be a sequence of harmonic functions that converge normally to $h$. Prove that $h_{n}'\rightarrow h'$ normally.

Certainly, this theorem holds when we have analytic functions (using Cauchy estimates). But how to generalize from here, using that every harmonic function is locally the real part of an analytic function, or Poisson integral...?

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    [All harmonic functions are analytic](http://en.wikipedia.org/wiki/Harmonic_function#Remarks).2011-08-23
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    Harmonic functions satisfy Cauchy estimates, almost by definition.2011-08-23

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