PROOF why Achilles never beats the Tortoise in a footrace.

Click for larger & clearer.

Okay, here's why Achilles, the fastest human on Earth, can't win a footrace against the Tortoise, the slowest animal on Earth.

Achilles and the Tortoise agree to have a foot race. Achilles is a sport, so he gives the Tortoise a generous head start of 10 meters (the Red length).

BANG!

Well, okay, Achilles can run 10 meters like greased lightning. That takes him 0.03 seconds.

But the Tortoise isn't dead, it's just Slow.
In the 0.03 seconds it took Achilles to run that small distance, the Tortoise crept forward 10 centimeters.

So now the Tortoise is 10 cm ahead of Achilles.

No problem. It only takes Achilles 0.0003 seconds to run that distance.

But the Tortoise isn't dead, it's just Slow.
In the 0.0003 seconds it took Achilles to run that small distance, the Tortoise crept forward 1 millimeter.

[from here on, distances are all in millimeters.]

So now the Tortoise is 1 mm ahead of Achilles.

No problem. It only takes Achilles 0.000003 seconds to run that distance.

But the Tortoise isn't dead, it's just Slow.
In the 0.000003 seconds it took Achilles to run that small distance, the Tortoise crept forward 0.01 millimeters.

So now the Tortoise is 0.01 mm ahead of Achilles.

No problem. It only takes Achilles 0.00000003 seconds to run that distance.

But the Tortoise isn't dead, it's just Slow.
In the 0.00000003 seconds it took Achilles to run that small distance, the Tortoise crept forward 0.0001 millimeters.

So now the Tortoise is 0.0001 mm ahead of Achilles.

No problem. It only takes Achilles 0.0000000003 seconds to run that distance.

But the Tortoise isn't dead, it's just Slow.
In the 0.0000000003 seconds it took Achilles to run that small distance, the Tortoise crept forward 0.000001 millimeters.

So now the Tortoise is 0.000001 mm ahead of Achilles.

No problem. It only takes Achilles 0.000000000003 seconds to run that distance.

But the Tortoise isn't dead, it's just Slow.
In the 0.000000000003 seconds it took Achilles to run that small distance, the Tortoise crept forward 0.00000001 millimeters.

So now the Tortoise is 0.00000001 mm ahead of Achilles ...

Obviously, Achilles will never overtake and pass the Tortoise. If he gives the Tortoise a head start, Achilles can never win the race. Which is what I set out to prove.