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.
Can anyone answer this question ??​

Answer 1

Answer:

You don't, because it's false. If all black dots happen to be on the line y=0 and white dots on the line y=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.

Answer 2

Answer:

This suggests that we color every point on the plane either white or black, making sure to use infinitely many of each color. With these constraints it is indeed the case that there must be a black point and a white point at unit distance. (In fact, "infinite" can be weakened to nonempty).