The Bolzano-Weierstrass Theorem says that any bounded sequence has a limit point. In other words points have to be as close as possible to some point. The applet demonstrates three cases of this. A bounded sequence can be uniform in distribution which is the random case in the applet. This results in the whole line being limit points. To see this choose random and then click on the line. Clicking on the line chooses a center to zoom into, epsilon chooses the radius of the interval. As can be seen all the points are close together no matter how much you zoom in. Convergent shows a random monotonic sequence and choosing epsilon lets you look at the interval from 1-epsilon to 1. The sequence is forced to converge to 1 because it keeps getting bigger but can't get any larger than 1. The applet shows this happening that no matter how you zoom in it always goes to 1. The third case is an infinite equally spaced sequence. You get limit points at the equally spaced points if you keep repeating them.

by the Coding Zebra