An Unanswered Question

OK, all you theoretical physicists out there:

I've been fascinated by the photos and information we've gotten from the Hubble Telescope. Not only are the pictures spectacular (look 'em up!) but the information gleaned about the size of the universe is truly mind-boggling. The very definition of it, in fact.

One piece of information from all those new discoveries has me puzzled. All those gazillion (technical term) galaxies are not only moving away from  us, it has been determined that the rate at which they are traveling is increasing! 

I heard that scientific speculation has therefore pushed the conclusion that eventually everything else will be so far away from us, that aside from our own galaxy, the night sky will be dark. Everything will be too far away to see, not because they are so tiny, but because their light can't possibly reach us. Now that's where I have a problem.

We believe, at this current moment (a qualifier given that knowledge historically increases and updates), that the speed of light is an absolute limit of possible movement. Nothing can travel faster. Nothing. So-o-o-o-o....

As all those other galaxies approach the speed of light, and we suppose their speed will match up with lightspeed, why will the sky go dark? Won't that light reach a - to us- static point where they will all appear to freeze in relation to us rather than winking out?

Yes, this is not an urgent question. My lifetime will be too short for this to be a remotely practical question. So will yours. The sun will likely burn itself out long before then, something that will likely prevent any earthly life from answering the question. In fact, even allowing for continually increasing advances in optics so one could actually follow retreating galaxies, solar burnouts would likely make it impossible for anything to answer that question, making it moot.

But just suppose....