The Tortoise Problem

Around 450 BC, the Greek philosopher Zeno proposed a puzzle: Achilles races a tortoise with a head start. By the time Achilles reaches the tortoise's starting point, the tortoise has moved ahead. Achilles reaches that point, but the tortoise moves again. Infinite steps — does Achilles ever pass?

Common sense says yes, Achilles obviously wins. But Zeno's logic seems airtight: there are infinitely many gaps to close. The resolution took over two thousand years. It required a rigorous understanding of infinity and the concept of a convergent series.