Every point of three-dimensional space is colored red, green or blue. Prove that one of the colors attains all distances, meaning that any positive real number represents the distance between two points of this color.
Let's suppose that no color attains all distances, meaning that exist real numbers $r \ge g \ge b$ such that no two red points have distance $r$, the same for g and green and b and blue.
Then consider a sphere of radius $r$ centered at a red point. Clearly its surface has green and blue points only.
Taking a look at the solution, they affirm that:
1) "since $g,b \le r$ the surface of the sphere must contain both green and blue points"
It doesn't seem clear to me and I don't know how to prove it. I think I must show that the points on the surface of the sphere of radius $r$ attains all distances between $0$ and $2r$ (and it would clearly imply that these points attains all distances between $0$ and $g$ or $b$).
2) "Let $M$ be a green point in the surface. There exist two points $P$ and $Q$ on the sphere such that $MP = MQ = g$ and $PQ = b$"
I also can't see how they can affirm that such points exist. With that two facts the problem is done.
I know that I should suppose the contrary to prove those things but even so I'm getting nowhere.