I haven't studied hyperbolic geometry in depth, but isn't limiting parallel a special case of parallel? I.e., if lines EF and GH are limiting parallel then they must also be parallel, but if lines IJ and KL are parallel, then they are not necessarily limiting parallel. I don't know the textbook definition of limiting parallel ray, but this page defines such a ray in terms of the associated lines, which would make your proof trivial.

Hope that helps..