1) The ball stops bouncing in a finite amount of time.
2) The ball bounces an infinite number of times.
3) This is not a contradiction, no more than the idea that a projectile passes through an infinite number of spatial points in finite time. Please go and read about geometric series and Zeno's paradox on wikipedia, as another commenter has already suggested.
This idea of "containment" may have worked well in the past, but is not as practical anymore. The world is so deeply connected, and technology so advanced, that international cooperation is far more essential.
Nuclear weaponry is but one of a number of salient examples. Even such staunch realists as Kissinger have come to recognize that they must be eliminated, and that intense international cooperation will be necessary to do so: