Consider the Bergman kernel $K_\Omega$ associated to a domain $\Omega \subseteq \mathbb C^n$. By the reproducing property, it is easy to show that $K_\Omega(z,\zeta) = \sum_{n=1}^\infty \varphi_k(z) \overline{\varphi_k(\zeta)},\qquad(z,\zeta\in\Omega)$ where $\{\varphi_k\}_{k=1}^\infty$ is any orthonormal basis of the Bergman space $A^2(\Omega)$ of Lebesgue square-integrable holomorphic functions on $\Omega$.
This series representation converges at least pointwise, since the Bergman kernel's Fourier series, $K_\Omega(\cdot,\zeta) = \sum_{k=1}^\infty \langle K_\Omega(\cdot,\zeta), \varphi_k \rangle \varphi_k$ with $\langle K_\Omega(\cdot,\zeta), \varphi_k \rangle = \overline{\varphi_k(\zeta)}$ converges in norm which implies uniform convergence in the first argument for fixed $\zeta \in \Omega$.
Now in Books such as Function Theory of Several Complex Variables by S. Krantz, it is shown that the series is uniformly bounded on compact sets, namely $ \sum_{k=1}^\infty \big| \varphi_k(z) \overline{\varphi_k(\zeta)} \big| \leq \bigg(\sum_{k=1}^\infty |\varphi_k(z)|^2 \bigg)^{1/2} \bigg(\sum_{k=1}^\infty |\varphi_k(\zeta)|^2 \bigg)^{1/2} \leq C(K)^2,\qquad(z,\zeta \in K)$ where $C(K)$ is a constant depending only on the compact set $K\subseteq \Omega$.
My question is this: Why does this imply uniform convergence on compact sets in $\Omega \times \Omega$? This is claimed in several sources, but just stated and not proven. Am I missing something obvious here?
One book which is a bit more specific is Holomorphic Functions and Integral Representations in Several Complex Variables by M. Range. There it is written that uniform convergence on compact subsets of $\Omega \times \Omega$ follows from the uniform bound and a "normality argument", which I take as referring to Montel's theorem. Does anyone know the details on how this argument works?
Any help is appreciated.