This article extends the results of Arkeryd and Cercignani[6].It is shown that the Cauchy problem for the relativistic Enskog equation in a periodic box has a global mild solution if the mass,energy and entropy of the initial data are finite.It is also found that the solutions of the relativistic Enskog equation weakly converge to the solutions of the relativistic Boltzmann equation in L1 if the diameter of the relativistic particle tends to zero.