ZENO'S PARADOX

The Tortoise challenged the great warrior Achilles to a 100 meter foot race, claiming that he would win as long as Achilles granted him a little headstart. Achilles laughed, for he was a mighty warrior swift of foot, whereas the Tortoise was heavy and slow.

"How long of a head start do you need?" asked Achilles, smiling.
"Ten meters," said the Tortoise.
Achilles laughs. "OK, you will most definitely lose, but we can race if you really want."
"Actually, I will most definitely win, and I can prove it to you with a simple argument," said the Tortoise.
"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.

Was it really impossible for Achilles to win the race? Explain.