Paradoxical life

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 feet. If we suppose that each racer starts running at some constant speed (one very fast and one very slow), then after some finite time, Achilles will have run 100 feet, bringing him to the tortoise's starting point. During this time, the tortoise has run a much shorter distance, for example 10 feet. It will then take Achilles some further time to run that distance, in which time the tortoise will have advanced farther; and then more time still to reach this third point, while the tortoise 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.

Paradoxes are fun. I try disproving this.

1) Achilles runs faster than the tortoise. Hence during any period of time, Achilles would cover more distance that the tortoise did and will eventually cancel out the initial difference in distance.

2) The paradox assumes that there is an infinite number of points Achilles must reach before catching up with the tortoise. However, this would also mean that it would take an infinite amount of time. Since time = distance / speed, therefore if infinity = d / x, then x = 0. This would mean that Achilles' speed is 0 which is illogical.