Prove that among any five points selected inside a square with side length 2 units, there always exists a pair of these points that are within $\sqrt{2}$ units of each other.

We can argue using the pigeon-hole principle...

Suppose we divide our square into 4 smaller square regions each of side length 1 unit, like so:

These smaller square regions will be our "pigeon-holes". Now, if one selects $5$ points (our "pigeons") inside the larger square, and noting that $5$ is one more than the number of smaller square regions -- the pigeon-hole principle requires that 2 of these points be in at least one of the smaller squares. Since the maximum distance between two points in one of these smaller squares is the length of the diagonal of that square, namely $\sqrt{2}$, we know there exists a pair of points out of the group of five originally selected that will be within $\sqrt{2}$ units of each other.