The existence and uniqueness of convex hull is trivial $\textbf{n -th Proposition : we cannot repeat P infinitely with n points on the plane}$ $\textbf{1st Proposition}$ is trivial and assume that $\textbf{k - th Proposition}$ $\forall k \in \mathbb{N} \cap [a,n-1]$ is true Assuming that $\textbf{n -th Proposition}$ is not true, Let the points consisting the convex hull of the $n$ points as $A_1,A_2,\cdots A_m$ $A_i$ cannot be a point which isn't one of the two edges of some line consisting three or more points Them the number written on $A_i$ is monotonously decreasing Hence, we can repeat $P$ consisting $A_i$ only finitely Therefore, we have to repeat $P$ infinitely without $A_i$ If we think of the points without $A_1$, there are $n-1$ points However, by $\textbf{n-1 - th Proposition}$ which we assumed true, we cannot repeat $P$ infinitely on $n-1$ points Therefore, by induction, the problem is solved