Zeno of Elea (~450 BC) argued that swift Achilles can NEVER catch a slow tortoise! Each time he reaches where it was, it has moved ahead. Infinite steps means infinite time... right?
Distance Achilles must travel:
Converges to: 11.111... = 100/9 meters
Partial sums approach the limit (dashed line)
Infinite steps, but FINITE time!
Zeno's paradox tricks us by conflating infinite steps with infinite time. But here's the key insight:
The infinite series of times also converges! Achilles catches the tortoise in just over one second. The "paradox" disappears once we understand that infinitely many things can happen in finite time if each takes proportionally less time.
Zeno of Elea created this paradox to defend his teacher Parmenides, who argued that all change and motion are illusions. The paradox remained philosophically troubling for over 2,000 years until Newton and Leibniz invented calculus in the 17th century, and Cauchy rigorously defined limits in the 19th century.
Other Zeno paradoxes include: The Dichotomy (must cross half the distance first, then half of that...), The Arrow (at any instant, an arrow is motionless), and The Stadium (relative motion paradox).