I need some help in proving the following lemma

Lemma . if $\displaystyle f_j $ are holomorphic functions on $\displaystyle U $ and $\displaystyle f_j \rightarrow $ uniformly on compact subsets of $\displaystyle U $, then any derivatives

$\displaystyle (\frac{\partial}{\partial z_1})^l (\frac{\partial}{\partial z_2})^k f_j $

converges to

$\displaystyle (\frac{\partial}{\partial z_1})^l (\frac{\partial}{\partial z_2})^k f $

Uniformly on compacts sets.

Proof:

i tried to fix $\displaystyle P \in U $ and choose $\displaystyle r > 0 $ such that $\displaystyle \overline{D}^2(P,r) \subseteq U$. i have some problem expressing $\displaystyle f_j on D^2(P,r) $ as a Cauchy integral of $\displaystyle f_j $ on $\displaystyle \partial D^2(P,r) $ and differentiating it under integral sign.

I need some help please