Zeno’s Paradox of Achilles and the Tortoise

Zeno of Elea (circa 450 b.c.) is credited with creating several famous paradoxes, but by far the best known is the paradox of the Tortoise and Achilles. (Achilles was the great Greek hero of Homer's The Iliad.) It has inspired many writers and thinkers through the ages, notably Lewis Carroll and Douglas Hofstadter, who also wrote dialogues involving the Tortoise and Achilles.
The original goes something like this:

            The Tortoise challenged Achilles to a race, claiming that he would win as long as Achilles gave him a small head start. Achilles laughed at this, for of course he was a mighty warrior and swift of foot, whereas the Tortoise was heavy and slow.
            “How big a head start do you need?” he asked the Tortoise with a smile.
            “Ten meters,” the latter replied.
Achilles laughed louder than ever. “You will surely lose, my friend, in that case,” he told the Tortoise, “but let us race, if you wish it.”
            “On the contrary,” said the Tortoise, “I will win, and I can prove it to you by a simple argument.”
            “Go on then,” Achilles replied, with less confidence than he felt before. He knew he was the superior athlete, but he also knew the Tortoise had the sharper wits, and he had lost many a bewildering argument with him before this.
            “Suppose,” began the Tortoise, “that you give me a 10-meter head start. Would you say that you could cover that 10 meters between us very quickly?”
            “Very quickly,” Achilles affirmed.
            “And in that time, how far should I have gone, do you think?”
            “Perhaps a meter – no more,” said Achilles after a moment's thought.
            “Very well,” replied the Tortoise, “so now there is a meter between us. And you would catch up that distance very quickly?”
            “Very quickly indeed!”
            “And yet, in that time I shall have gone a little way farther, so that now you must catch that distance up, yes?”

            “Ye-es,” said Achilles slowly.
            “And while you are doing so, I shall have gone a little way farther, so that you must then catch up the new distance,” the Tortoise continued smoothly.
Achilles said nothing.
            “And so you see, in each moment you must be catching up the distance between us, and yet I – at the same time – will be adding a new distance, however small, for you to catch up again.”
            “Indeed, it must be so,” said Achilles wearily.
            “And so you can never catch up,” the Tortoise concluded sympathetically.
            “You are right, as always,” said Achilles sadly – and conceded the race.