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

Prove that there is a sequence of points $\displaystyle \left( p_n \right)_{n \geq 1}$ such that $\displaystyle p_n \in P_n$  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?

[i]Claudiu Raicu[/i]
1. **Define the Graph:**
   Consider a graph \( G \) where the vertices are the elements of \( \bigcup_{n \geq 1} X_n \). A pair of points \( \{p_n \in X_n, p_{n+1} \in X_{n+1}\} \) is connected by an edge if \( p_{n+1} \) projects orthogonally onto \( p_n \).

2. **Finite Degree:**
   Each vertex in this graph has a finite degree because each \( X_n \) is a finite set, and the projection of \( X_{n+1} \) onto \( P_n \) is included in \( X_n \).

3. **Connected Components:**
   Each connected component of the graph contains at least one element from \( X_1 \). Since \( \left( X_n \right)_{n \geq 1} \) is an infinite family, there must be at least one connected component that is infinite.

4. **Application of Konig's Infinity Lemma:**
   Konig's Infinity Lemma states that in a connected graph where every vertex has finite degree, and given a distinguished vertex \( p_1 \), there exists an infinite simple path starting at \( p_1 \). Apply this lemma to the infinite connected component found above, with \( p_1 \) being one of its vertices in \( X_1 \).

5. **Constructing the Sequence:**
   By Konig's Infinity Lemma, there exists an infinite simple path \( (p_n)_{n \geq 1} \) such that \( p_n \in P_n \) and \( p_n \) is the projection of \( p_{n+1} \) onto \( P_n \).

Thus, we have constructed the required sequence \( (p_n)_{n \geq 1} \).

6. **Counterexample for Infinite Sets:**
   Consider the planes \( P_n = \{(x, y, z) \in \mathbb{R}^3 \mid z = n\} \) and the sets \( X_n = \{(k, 0, n) \mid k \geq n\} \). Here, \( X_n \) are infinite sets. The projection of \( X_{n+1} \) onto \( P_n \) is \( \{(k, 0, n) \mid k \geq n+1\} \), which is included in \( X_n \). However, there is no sequence \( (p_n)_{n \geq 1} \) such that \( p_n \in P_n \) and \( p_n \) is the projection of \( p_{n+1} \) onto \( P_n \) for all \( n \).

The final answer is True for finite sets \( \boxed{ X_n } \) and False for infinite sets \( X_n \).