Let
![n \geq 2, n \in \mathbb{N}](/media/m/9/9/0/9902fc58bce61021cef06d5728b5a1fd.png)
and
![A_0 = (a_{01},a_{02}, \ldots, a_{0n})](/media/m/3/6/c/36c6b6ed77a27a2e2b8bc7472300d765.png)
be any
![n-](/media/m/f/a/e/fae563323c2368fde7e704b858164853.png)
tuple of natural numbers, such that
![0 \leq a_{0i} \leq i-1,](/media/m/4/0/7/4072bcd330a998a5857a8bdb15fcc6b7.png)
for
![n-](/media/m/f/a/e/fae563323c2368fde7e704b858164853.png)
tuples
![A_1= (a_{11},a_{12}, \ldots, a_{1n}), A_2 = (a_{21},a_{22}, \ldots, a_{2n}), \ldots](/media/m/6/8/5/6859298770a6310a3daf81fec33f78c4.png)
are defined by:
![a_{i+1,j} = Card \{a_{i,l}| 1 \leq l \leq j-1, a_{i,l} \geq a_{i,j}\},](/media/m/0/2/8/0286b43d9a130ccc8c9b301c50dee7b0.png)
for
![i \in \mathbb{N}](/media/m/9/8/1/98124ab61852dc52c8b314172f83fc91.png)
and
![j = 1, \ldots, n.](/media/m/5/5/8/558a2ef32e3415815b822c57bdb57b11.png)
Prove that there exists
![k \in \mathbb{N},](/media/m/7/0/a/70aef2bc416ab2b9708319d36c05f2b0.png)
such that
%V0
Let $n \geq 2, n \in \mathbb{N}$ and $A_0 = (a_{01},a_{02}, \ldots, a_{0n})$ be any $n-$tuple of natural numbers, such that $0 \leq a_{0i} \leq i-1,$ for $i = 1, \ldots, n.$
$n-$tuples $A_1= (a_{11},a_{12}, \ldots, a_{1n}), A_2 = (a_{21},a_{22}, \ldots, a_{2n}), \ldots$ are defined by: $a_{i+1,j} = Card \{a_{i,l}| 1 \leq l \leq j-1, a_{i,l} \geq a_{i,j}\},$ for $i \in \mathbb{N}$ and $j = 1, \ldots, n.$ Prove that there exists $k \in \mathbb{N},$ such that $A_{k+2} = A_{k}.$