Zeno's paradoxes
In the paradox of Achilles and the Tortoise, Achilles is in a footrace with the tortoise. Achilles allows the tortoise a head start of 100 meters, for example. Suppose that each racer starts running at a constant speed, one very fast and one very slow. After some finite time, Achilles will have run 100 meters, bringing him to the tortoise's starting point. During this time, the slower tortoise has run a much shorter distance. It will then take Achilles some further time to run that distance, by which time the tortoise will have advanced farther. It will then take still more time for Achilles to reach this third point, while the tortoise again moves ahead. Thus, whenever Achilles reaches somewhere the tortoise has been, he still has farther to go. Therefore, because there are an infinite number of points Achilles must reach where the tortoise has already been, he can never overtake the tortoise.