Let $D subset C^n$ be a smoothly bounded pseudoconvex domain and let ${ar{D}_r}_r$ be a family of smooth perturbations of $ar{D}$ such that $ar{D} subset ar{D}_r$. Let $K_D(z, w)$ be the Bergman kernel function on $D imes D$. Then $lim_{r o 0} K_{D_r}(z, w) = K_D(z, w)$ locally uniformally on $D imes D$.