If C is a chain of set S, $C \subseteq S$ or $C \subseteq S \times S$?

Question

I'm doing an exercise, which asks us to prove that "every infinite set $S \subseteq \mathbb N^n_0$ contains an infinite chain $C\subseteq S$." But since C is a chain of set S, C is supposed to be a relation on S, which means, suppose P is a poset of S, $P=(S,\preceq)$, $P \subseteq S \times S$, and $C \subseteq P$, so actually $C \subseteq S \times S$. Or maybe in this context it's also understandable to write $C \subseteq S$? Could anyone please tell me what's wrong here? I think I've misunderstood some concepts. Thank you very much.

Answer 1

No, $C$ is not supposed to be a relation on $S$. I guess that the relation $\le$ on $S$ is induced from $\Bbb N_0$ and $x=(x_1,\dots,x_n)\le (y_1,\dots,y_n)=y$ iff $x_i\le y_i$ for each $i$.