Achilles Paradox

Paradox
Achilles and a tortoise are to have a race.

Achilles, not surprisingly, runs considerably faster than the tortoise. Therefore, he gives the tortoise a head start in order to make the race more fair.

However, it is apparent that Achilles will not be able to catch up with the tortoise.

Suppose he gives the tortoise a headstart of $x_0$m.

By the time he has run $x_0$m, the tortoise has moved on, by $x_1$m, say.

But by the time Achilles has run another $x_1$m, the tortoise has moved on, by $x_2$m, say.

And this can be continued indefinitely, so Achilles will never catch up with the tortoise.

Resolution
It is clear that there is a problem with this reasoning, as it is tantamountly clear that someone running faster than another will overtake, sooner or later.

The solution depends on the concept of a limit.

The sum of the distances run by Achilles in catching up the tortoise is an infinite series which is bounded above.

As such, once Achilles reaches that limit, any further distance he travels will bring him further than the tortoise.

The problem lies in the assumption the Achilles is bounded to points only previously set by the tortoise, however in all practicality this is not the case (Achilles' step size).

Origin
This paradox was famously raised by Zeno of Elea.