Hello, JoshuaJava!

Here is a rather juvenile approach.

Q: Show that if 5 points are placed in a unit square.

then there are two points no more than units apart.

Place *four* points in a square with maximum distances between them.

. . They will be at the *vertices* of the square.

Code:

♥ - - - - - ♥
| |
| |
1 | |
| |
| |
♥ - - - - - ♥
1

Now place a fifth point in the square,

. . maximizing its distances from the first four points.

Code:

♥ - - - - - ♥
| * * |
| * * |
1 | ♠ |
| * * |
| * * |
♥ - - - - - ♥
1

The best we do is use the *center* of the square.

And we find that its distance from its neighbors is