If you drop a bouncy ball on a level surface doesn’t it never stop bouncing? I realized this because of the Infinite Divisibility Theory. Which states that you can infinitely keep splitting something in half. Furthermore it would keep bouncing because there would always be a space between the surface and the ball’s surface. Also the only reason why we can’t see it bouncing is because it is an extremely small distance between the surface of the ball and the surface it’s bouncing on.
 A: To answer this question you have to distinguish between the physical system and the mathematical model.
If the bounces are from inelastic collisions between the ball and the immovable surface then the bounce height never decreases and the motion is periodic. In the real world that's impossible.
For elastic collisions you need to take into account the energy lost to heat when the ball and the surface momentarily contract and rebound at the bounce. Then the bounces are smaller and smaller as time goes by. I can imagine mathematical models where the magnitude decays to 0 in finite time  or in infinite time (I don't know enough physics to decide which is realistic. I suspect the former).
But any good model of the macroscopic behavior will fail at small times and distances. Then quantum mechanics takes over.
A: Mathematics is a tool for modelling real world phenomena; real world phenomena is not mathematics. Math will only be as useful insofar as the asssumptions underlying the math are dually satisfied by the real world system. For instance, you reasoning suggests that space can be infinitely divided, but there is a limit to the smallest unit of space known as the Planck distance, according to the best currently accepted theory in physics.
