## History and applications

The ancient Greek philosophers appear to have been the first to contemplate the infinite in a formal way. The concept worried them somewhat, and Zeno came up with a number of paradoxes which they were not really able to explain properly. Here are two of them:

Suppose I travel from $$A$$ to $$B$$ along a straight line. In order to reach $$B$$, I must first travel half the distance $$AB_1$$ of $$AB$$. But to reach $$B_1$$ I must first travel half the distance $$AB_2$$ of $$AB_1$$, and so on ad infinitum. They then concluded that motion is impossible since, presumably, it is not possible to complete an infinite number of tasks.
A tortoise is racing against Achilles and is given a head start. Achilles is much faster than the tortoise, but in order to catch the tortoise he must reach the point $$P_1$$ where the tortoise started, but in the meantime the tortoise has moved to a point $$P_2$$ ahead of $$P_1$$. Then when Achilles has reached $$P_2$$ the tortoise has again moved ahead to $$P_3$$. So on ad infinitum, and so even though Achilles is faster, he cannot catch the tortoise.