But I think that's the whole issue-- I think that's what Zeno was concerned with. Even if one knows the mathematics of infinite sums, as Ilya correctly pointed out, you still need time to be perfectly subdividable to use those mathematics. We pretty much already know that time is not so subdividable (the uncertainty principle). My point is that the basic error is in trying to use a concept, like time, to learn about how reality works. That's backward-- we use reality to learn about how we need to make a functioning concept of time, and the function we achieve should never be expected to extend beyond where we have measured it. So Achilles does catch the tortoise, and the mathematics of infinite sums tells us just how long it will take, but it is not necessary for the details of what is happening be taken too literally-- mathematics applied to the real world is just a tool, it is not the same thing as the real world.