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.

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Answer 1

Answer:

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).

Answer 2

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There are infinite black and white dots on a plane. ... In either case we can see that plane is colored with black or white color only. So we can say that for each white dot there is exist a black dot at unit distance because if not than we end up coloring the full plane with white color.

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