Question

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

Answer 1

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.