Zeno's paradox is an interesting mathematical construct, but does it really exist in the real world? Do you really need infinity to describe motion? Only in an infinite universe. A finite universe doesn't require infinity to describe motion. Think of it like pixels in a video game. It doesn't make sense to say that there are an infinite amount of pixels between two pixels, because you know there are a finite amount of pixels on your screen (1920x1080 for example). But yet you see motion in your video game. Why? Motion is an illusion. There are only instantaneous state changes, that occur so rapidly and at such small scale our brains interpret them as the abstract concept of motion.

Our brains are also finite and therefore can only process a finite amount of information in a given period of time. However we want to draw conclusions and make predictions even though we only ever observe a tiny bit of the information thrown at us. We can use mathematics to make assumptions about the information we never directly observed. This proves helpful in most situations, but could also lead us to incorrect conclusions.

Could the real universe be the same way? There are a finite amount of atoms between two points on a sheet of paper.
Are there a finite amount of particles within an atom? An electron? A quark? No one can say for sure... yet.

I'm not a physicist, but it is interesting that the wiki page http://en.wikipedia.org/wiki/Quark does not ever use the words infinity or infinite.

So is infinity a mathematical construct, or does it really exist?