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

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.

@WhizKidXAman☺✅

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.