Brandon Watson has a discussion on Zeno's Paradox here. He refers to a version of the argument given at the Internet Enyclopedia, but I don't follow it. The salient points are listed below. I can see that 1-4 are true. But why is 5 true? How are 1-4 supposed to imply 5?
1. The tortoise has a head start, so if Achilles hopes to overtake it, he must run at least to the place where the tortoise presently is
2. But by the time he arrives there, it will have crawled to a new place.
3. So then Achilles must run to this new place.
4. But the tortoise meanwhile will have crawled on, and so forth.
5. Achilles will never catch the tortoise, says Zeno.