Achilles Paradox

Paradox
Achilles and a tortoise are to have a race.

Achilles (not surprisingly) runs considerably faster than the tortoise. So, to make it marginally more fair, he gives the tortoise a head start.

But it is apparent that Achilles can not actually catch up with the tortoise.

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

By the time he has got to $$x_0$$, the tortoise has moved on, to $$x_1$$, say.

But by the time Achilles has reached $$x_1$$, the tortoise has moved on, to $$x_2$$, say.

You can continue this indefinitely.

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.

Origin
This paradox was famously raised by Zeno of Elea.