How do we know that the galaxies the farthest away from us are actually speeding up? What i mean is Hubble proved that the galaxies the farthest away from us were moving away faster than the galaxies closer to us. I have a basic understanding of red shift. I've always wondered (and have never found an answer) how do we know that its a product of distance and not a product of time. How do we know that expansion isn't actually slowing down and that what we are actually seeing is how fast galaxies were moving all those billions of years ago and the closer we get to our time (closer galaxies) the slower the expansion is getting. Telescopes are time machines. We are looking back just as much as we are looking out. Ive always been curious about that. Its not like we can see what a galaxy at the edge of the observable universe is doing right now. Thanks for taking time to answer a layman's question. 
 A: The cosmological redshift of a galaxy is not interpreted as being due to the velocity of that galaxy away from us (the special relativistic interpretation), but rather as being being due the effect of the expanding space on the traveling photon (the general-relativistic interpretation).$^\dagger$
This expansion in turn makes the galaxies recede from us at a velocity that increases with their distance from us.
If you consider the redshift of a single galaxy, in principle you cannot know whether space has been expanding continuously between the time $t_\mathrm{em}$ of emission and the time $t_\mathrm{obs}$ of observation, or whether it were static at $t_\mathrm{em}$ and static at $t_\mathrm{obs}$, but had a period of expansion in between. All the redshift tells you is the ratio of the size (or the "scale factor" $a$) of the Universe then and now.
However, when we look at all galaxies, we see that the redshift increases with their distance from us. The distance can be translated into a time, considering the time the light has spent traveling that distance. That means that we get a picture of the size of the Universe as a function of its age. And that size has not only been monotonically increasing ever since Big Bang, but we see that it increased at a slower and slower rate in the beginning, but then at some point started increasing faster and faster.

$\dagger$The SR interpretation is in principle a valid possibility, but can be ruled out at 23$\sigma$ (Davis & Lineweaver 2004).
A: You say right as Hubble's law state that farthest the galaxy is faster it is receding from us.
$v=Hl$. H is Hubble constant. $H(t)=\frac{\dot{a(t)}}{a(t)}$, a(t) is scale factor. Experimentally it is confirmed by observing the radiation from distant object. Due to Doppler's effect observed frequency of light will be different from emitted one and it depends upon receding velocity. we compare the observed spectrum with our standard laboratory data and find the velocity of object. Comparing the velocity result our long time we find that whether it is accelerating or slowing down. There are other observational technique also for observing the expansion of universe.
