This video illustrates a proof of the Heine Borel Theorem: "Every closed bounded set in R^n is compact."
The proof goes like this:
Assume the contary. Let X be a bounded closed set (the disk in the video) that has an infinite open covering (symbolized by the colored small disks in the video) with no finite subcovering. Now devide X and the covering in two halfs. Now at least one sides does not admit a finite subcovering (otherwise we could construct a finite subcovering of X). Now subdivide the side without finite subcovering, and repeat the construction. We obain a sequence of subdivisions that do not have a finite subcovering of the given infinite covering.
If we now choose a point inside each of the subdivisions we obtain a cauchy sequence, which must have a limit point P since R^n is complete. P lies in every subdivision. Now choose an open set U (yellow in the movie) of the covering which contains P. Since U has a finite size it contains all subdivisions which are small enough. But then U is a finite subcovering for these subdivisions which contradicts our construction. Therefore our assumption that allowed the construction must be wrong and the Heine-Borel-Theorem correct. q.e.d
This Video was produces for a topology seminar at the Leibniz Universitaet Hannover.