Convergence of Invariant Distances on Decreasing Domains

We investigate the behavior of the invariant distances and metrics on decreasing domains in complex Euclidean space. The invariant distances on a domain with some conditions are the limit of them on decreasing sequence of domains converging to the domain. The conditions are the strong pseudo-convexity with smooth boundary for the Carathéodory distance and the existence of weak peak functions for the Kobayashi one.