SCHWEITZER 1992 Problem 10

We place n points in the unit square independently, according to a uniform distribution. These points are the vertices of a graph $G_n$ . Two points are connected by an edge if the slope of the segment connecting them is nonnegative. Denote by $M_n$ the event that the graph $G_n$ has a 1-factor. Prove that $\lim_{n \to \infty} P(M_ {2n}) = 1$ .