Question

There are infinite black and white dots on a plane. Prove that the distance between one black dot and one white dot is one unit.​

Answer 1

Answer:

How do you prove that the distance between one black dot and one white dot is one unit ? You don't, because it's false. If all black dots happen to be on the line and white dots on the line (and the rest of the plane is neither white nor black), there is no such pair.

Answer 2

Explanation:

starting with a black point, we're done unless the entire unit circle around that point is black. Now repeat that argument for each point on that unit circle: we've already generated a sizable swathe of the plane colored totally black...

The statement is in general false: for example, if the black dots are all and only on the locus x=0, while the white ones are all and only on the locus x=2, then the assertion is false (we are using the Euclidean distance).

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