RO-NMO 2004 Problem 104

Let $\left( P_n \right)_{n \geq 1}$ be an infinite family of planes and $\left( X_n \right)_{n \geq 1}$ be a family of non-void, finite sets of points such that $X_n \subset P_n$ and the projection of the set $X_{n+1}$ on the plane $P_n$ is included in the set $X_n$ , for all $n$ .

Prove that there is a sequence of points $\left( p_n \right)_{n \geq 1}$ such that $p_n \in P_n$ and $p_n$ is the projection of $p_{n+1}$ on the plane $P_n$ , for all $n$ .

Does the conclusion of the problem remain true if the sets $X_n$ are infinite?

Claudiu Raicu