In the proof of Caratheodory's lemma, why do we stop at $d+1$ points?

Question

I have to proof caratheodorys lemma for my oral exam. The proof is given here. I dont get the last part. "This process can be repeated until x is represented as a convex combination of at most d + 1 points in P" Why do we have to stop at d+1 points ? An explanation of that would be really appriciated:)

Answer 1

As the proof states, for $k > d+1$, the $k-1$ points $x_2 − x_1, \ldots, x_k − x_1$ must be linearly dependent, since they are in a $d$ dimensional space.

For $k \le d+1$, it is possible that the $k-1$ points $x_2 − x_1, \ldots, x_k − x_1$ are linearly independent, so it is not necessarily true that the process of eliminating points can continue.