How fast is the universe expanding? I've hear several theories stating that the universe is expanding faster than the speed of light, others claim that the universe expands faster the further away you measure it. Which of this is correct and how do you prove it (mathematically)? Furthermore, does this correctly imply, then, that eventually galaxies will be so far away, and moving so fast, that we will never see them again?

Thank you in advance!
Both are correct, although the first can be further explained a bit. I won't give you a mathematical proof, though; Instead I'll play with characters.

Let's assume, for the sake of gedankenexperiment, that:


The speed of light is 5 characters/sec;
our universe is expanding at 1 character per 5 characters per sec.


This is our current universe, and we launch a photon from body A aiming at body E.
(Space generated each second is marked with a # symbol.)

T=0s A----B----C----D----E           Bodies
     *                               Photon     - 19 chars to E
T=1s A--#--B--#--C--#--D--#--E 
      --#--*                                      17 chars to E
T=2s A--#---B-#----C#-----#D----#-E 
      --#-----#-*                                 17 chars to E
T=3s A--#----B#-----#-C---#----D#-----#-E 
      --#-----#-----#*                            18 chars to E
T=4s A--#-----#B----#----C#-----#---D-#-----#--E 
      --#-----#-----#-----#*                      19 chars to E


I know, the graph isn't too granular, and the space generation isn't evenly distribute. I apologize for that, but it's for the sake of demonstration.

Notice that at T=2 some space is already generated between A and the photon. But that's irrelevant: E is sitting at the event horizon, and will never be reached by the photon, because the amount of space being generated between photon * and body E is equal, or superior, to the speed of light.

Given any positive expansion rate, there will be an event horizon - a point where the accumulated dilation of space is more than the amount of space a particle moving at the speed of light can travel.

A galaxy sitting initially at say, 1000 characters from A at T=0, will be at staggering 1200C at T=1 - that's 40 times our speed of light.

At T=16s, B (that was passed by the original photon at T=1) will be sitting exactly where E was relative to A, and at T=17 will fall out of our event horizon. A new photon emitted from A will never reach it.