Heine-Borel theorem

For those who are interested in one fun "proof" of what we do in the school.
Heine-Borel theorem on Youtube.

What does it state?

Heine-Borel theorem: Let there be given an interval (a, b) and a set of intervals I each the members of which is in (a, b). Moreover, let I possess the properties:

(i) Every point of (a, b), other than a or b, lies inside, i.e., 'in and not at an end of', at least one interval of I;

(ii) a is the left hand and b the right hand end point of at least one interval of I.

Then it is possible to choose a finite number of intervals from the set I which form a set of intervals with the properties (i) and (ii).

Note that the set of intervals may have well been infinite and still, you can pick just a finite number while retaining all the "information".