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:

In my opinion, for this if we just imagine the scenario we comes up with a picture with "infinite white dots on black background" or vice versa . 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.

Answer 2

In My Opinion This Is False⤵

If all black dots happen to be on the line x=0and white dots on the line x=π (and the rest of the plane is neither white nor black), there is no such pair.

If all black dots happen to be on the line x=0and white dots on the line x=π (and the rest of the plane is neither white nor black), there is no such pair.Now if each point of the plane were either black or white (and there were infinitely many of each type), that would be different. In fact, it is sufficient to have at least one of each color.