SCHWEITZER 1978 Problem 8

Let $X_1, \ldots ,X_n$ be $n$ points in the unit square ( $n>1$ ). Let $r_i$ be the distance of $X_i$ from the nearest point (other than $X_i$ ). Prove that the inequality $$r_1^2+ \ldots +r_n^2 \leq 4.$$

L. Fejes-Toth, E. Szemeredi