This article is from the Puzzles FAQ, by Chris Cole firstname.lastname@example.org and Matthew Daly email@example.com with numerous contributions by others.
A bug walks down a rubber band which is attached to a wall at one end and a car
moving away from the wall at the other end. The car is moving at 1 m/sec while
the bug is only moving at 1 cm/sec. Assuming the rubber band is uniformly and
infinitely elastic, will the bug ever reach the car?
Let w = speed of bug and N = ratio of car speed/bug speed = 100. Paint N+1
equally spaced stripes on the rubberband. When the bug is standing on one
stripe, the next stripe is moving away from him at a speed slightly < w
(relative to him). Since he is walking at w, clearly the bug can reach
the next stripe. But once he reaches that stripe, the next one is only
receeding at < w. So he walks on down to the car, one stripe at a time.
The bug starts gaining on the car when he is at the next to last stripe.