The notion of "proof" in a field like mathematics simply doesn't exist in physics. In mathematics, certain "facts" can be definitively shown to be true. Such is not the case in physics or any other science; it is fundamentally against the philosophy of science.
"The test of all knowledge is experiment. Experiment is the sole judge of scientific 'truth'". -Richard Feynman
Einstein arrived at his theory of special relativity by guessing
that the speed of light is constant in all inertial frames. He did not "prove" it with mathematics anywhere, that would be impossible.
From this assumption, Einstein was able to form his theory of special relativity. In addition to a constant speed of light, special relativity predicts other phenomenon that can be tested with experiments. Additionally, the groundwork of special relativity has enabled the development of nearly every modern theory of physics, predicting all sorts phenomenon that Einstein could have scarcely imagined in 1905.
Why do we readily accept a constant speed of light as "truth"? Simply because, as far as we can tell, it seems to be true. We've never been able to perform an experiment that conclusively shows that the speed of light isn't constant in every inertial frame. Nor have we been able to conduct an experiment that contradicts the other predictions from Einstein's theory (in a weak gravitational field). This is what some would call "empirical proof" (empirical is just a fancy word describing information found from evidence and observation), though I've never been fond of that term, since it isn't a proof at all.
If we ever do find good evidence that contradicts a constant speed of light, then we would no longer accept the notion to be true. This is how science progresses. Before Einstein came along the world did mostly fine with classical Galilean relativity. And, as far as they could tell, it was the truth! Only as our experiments became more advanced were we able to see that it only holds true given constraints, and that new theories were needed.
It's worth noting that just because a theory isn't exactly consistent with modern experiments, doesn't mean it's not useful. For example, many things we deal with on earth can be dealt with just fine using classical mechanics, not accounting for relativistic effects. We know that it is technically not the best model we have, but the effect is so small that we can get along just fine with our old models. However, in systems like the Global Positioning System, we cannot get away with using old physics. When we want to model things that are more "extreme" (e.g. satellites orbiting the earth) we must use our more accurate models.
(published on 02/03/2012)