US-TST 2000 Problem 5

Let $n$ be a positive integer. A $corner$ is a finite set $S$ of ordered $n$ -tuples of positive integers such that if $a_1, a_2, \ldots, a_n, b_1, b_2, \ldots, b_n$ are positive integers with $a_k \geq b_k$ for $k = 1, 2, \ldots, n$ and $(a_1, a_2, \ldots, a_n) \in S$ , then $(b_1, b_2, \ldots, b_n) \in S$ . Prove that among any infinite collection of corners, there exist two corners, one of which is a subset of the other one.