Let $Omega$ be a bounded pseudoconvex domain in$C^{n}$ with smooth defining function r and let$z_0; {in}; b{Omega}$ be a point of finite type. We also assume that $Omega$ is convex in a neighborhood of $z_0$. Then we prove that all the holomorphic sectional curvatures of the Bergman metric of $Omega$ are bounded below by a negative constant near $z_0$.