We prove a global existence theorem for the $3\times 3$ system of relativistic
Euler equations in one spacial dimension. It is shown that in the ultra-relativistic limit,
there is a family of equations of state that satisfy the second law of thermodynamics for
which solutions exist globally. With this limit and equation of state, which includes
equations of state for both an ideal gas and one dominated by radiation, the relativistic
Euler equations can be analyzed by a Nishida-type method leading to a large data existence
theorem, including the entropy and particle number evolution, using a Glimm scheme. Our
analysis uses the fact that the equations of state are of the form $p=p(n,S)$, but whose
form simplifies to $p=a^{2}\rho$ when viewed as a function of $\rho$ alone.