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

How do you prove that the distance between one black dot and one white dot is one unit ? 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.

Explanation:

d.) continue to rise rapidly