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 $\displaystyle \tfrac{1}{\sqrt{2}}$ units apart.

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

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

Code:

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1 | |
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Now place a fifth point in the square,

. . maximizing its distances from the first four points.

Code:

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1 | ♠ |
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The best we do is use the *center* of the square.

And we find that its distance from its neighbors is $\displaystyle \frac{1}{\sqrt{2}}$