Zeno's Paradox: Achilles and the Tortoise

Zeno of Elea (circa 450 BC) is credited with creating several famous paradoxes, but by far the best known is the paradox of the tortoise and Achilles.

A fleet-of-foot Achilles is unable to catch a plodding tortoise which has been given a head start, since during the time it takes Achilles to catch up to a given position, the tortoise has moved forward some distance. But this is obviously fallacious since Achilles will clearly pass the tortoise!