Well, that part about “infinite mass” is a limit. Unless you want to reverse the Big Bang, or whatever.
Looked at from the other direction .... I recently read that Einstein puzzled over "catching up" to a photon. If an observer is traveling at the same speed and in the same direction as a photon, the photon (which is massless) would be standing still. And, if it is standing still and has no mass, there is nothing there. So, he figured, it must not be possible to observe a stationary photon.