Question 16
(MCD 3) Let $n\in\mathbb N$, $n\ge2$, and $A_0=(a_{01},a_{02},\ldots,a_{0n})$ be any $n$-tuple of natural numbers such that $0\le a_{0i}\le i-1$, for $i=1,\ldots,n$. The $n$-tuples $A_1=(a_{11},a_{12},\ldots,a_{1n})$, $A_2=(a_{21},a_{22},\ldots,a_{2n}),\ldots$ are defined by
$$a_{i+1,j}=\operatorname{Card}\{a_{i,l}\mid1\le l\le j-1,\ a_{i,l}\ge a_{i,j}\},\qquad i\in\mathbb N,\ j=1,\ldots,n.$$
Prove that there exists $k\in\mathbb N$ such that $A_{k+2}=A_k$.
$$a_{i+1,j}=\operatorname{Card}\{a_{i,l}\mid1\le l\le j-1,\ a_{i,l}\ge a_{i,j}\},\qquad i\in\mathbb N,\ j=1,\ldots,n.$$
Prove that there exists $k\in\mathbb N$ such that $A_{k+2}=A_k$.