I think the title says it all.
If you have a sequence of harmonic functions from a bounded complex domain to the real numbers, show that on a subset at a positive distance from the boundary of the domain, e.g. a compact subset of the domain, the derivatives of the harmonic functions converge uniformly to the derivative of the limit of the sequence of the harmonic functions.
My attempt: I tried to apply the mean value property (like Gamelin does for analytic functions) to find a bound (unsuccessfully). I know the question has been asked before, but I did not understand the solution. I also tried to come up with something similar to Cauchy estimates for harmonic functions, but I wound up more confused than when I started.
EDIT: Keep in mind by derivative I meant partial derivatives. My attempt was to find an analogue of the Cauchy integral formula (for derivatives) but I seem to get the same formula for both partial derivatives, which does not make sense to me because it seems like the partial derivatives can be different.