A question related to Extremal Principle

Question

Let $\omega$ be the set of points in plane. Each point in $\omega$ is a midpoint of two points in $\omega$. Show that $\omega$ is an infinite set.

Answer 1

We consider all points in $\omega$ farthest to the left, and among those the point $M$ farthest down. $M$ cannot be midpoint of the two points $$A, B\in\omega$$ since one element of $(A,B)$ would be either left of $M$ or on vertical below of $M$.