The Vanishing Lead

Achilles races a tortoise along a straight track. Because Achilles runs much faster, he allows the tortoise a head start of 100 metres. To overtake it, Achilles must first reach the tortoise’s starting point. During that time, the tortoise moves a little farther. Achilles must then cover that new gap, but while he does, the tortoise advances again. Repeating the argument creates an endless sequence of smaller gaps: whenever Achilles arrives at a place the tortoise previously occupied, it has already crept beyond it. Since there always seems to be another gap left to close, does Achilles ever pass the tortoise?