The aim of this work is to establish a proof that the speed of light in vacuum has to be infinite for certain physical problems. In the problem taken up there exist two simple media with finite conductivities, one of which is in uniform rectilinear motion with respect to the other. The media have an infinite plane boundary perpendicular to the direction of the motion. The finiteness of the conductivities precludes existence of induced surface current densities on the boundary. Additionally the two media are assumed to have equal relaxation times when both are at rest. Equal relaxation times for the two media have the effect of a zero induced charge density on the interface of the media. This is shown to be true both when the two media are at rest and when one is in uniform rectilinear motion. Then the boundary conditions observed from the rest frame are found to yield the infinite speed of light in vacuum.