Abstract: Weak convergence of Markov jump processes is characterized by the convergence of their imbedded chains. Especially, convergence theorem of homogeneous Markov jump processes is characterized by their infinitesimal characteristics and discrete approximation of homogeneous Markov jump processes are obtained. Since weak convergence of stochastic sequences needs only the convergence of finite-dimensional distributions, we don’t use general results on convergence of stochastic processes or Markov processes. The discussion is direct and elementary.