We all know that the Hubble equations tell us that in a vacuum dominated universe, that the scale factor will expand exponentially. H^2 = (da/dt)^2 * 1/a^2 Ha = da/dt a = e^(Ht) Where proper distance is d = aΔx v = Hd v = HaΔx <- The velocity between any two points in space is proportional to a soon to be infinite scale factor. Where x is just coordinate convention. In a vacuum dominated universe, H is constant due to the vacuum density being constant. As a t approaches infinity, so does the scale factor (a). Therefore, in the far distant future our universe will exponentially grow to infinity no matter the coordinate distance, since proper distance is proportional the scale factor. Does this not mathematically prove that the big rip is true?