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:

you don't because it is false. if all black dot happened to beyond the line of white dots on the line there is no such pair..

Answer 2

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.

Why? Pick any two points A and B that have different colors. Starting at A , we can reach B using a finite number of steps, each of length exactly 1: just go directly towards B until the distance becomes less than 1, and at the end, if we didn't reach B exactly, we make two steps "to the side and back" to reach it. (Formally, if you are currently at C , imagine circles with radius 1 centered at B and C . Pick one of their two intersections, go from C to that intersection and from there to B .)