Proof of Dickson's Lemma

You are almost there, you need only to repeat this for all the coordinates (note that this would fail if your tuples were of infinite size).

As you have observed, any infinite sequence of natural numbers $(a_n)_{n \in \mathbb N} \subseteq \mathbb{N}$ contains an infinite non-decreasing subsequence, in particular, either there is an element $k \in \mathbb{N}$ that happens infinitely many times, or there is an infinite increasing subsequence.

Let $F$ be any function $F : \mathbb{N}^\mathbb{N} \to \mathbb{N}^\mathbb{N}$ that extracts indices of some subsequence with the aforementioned property, that is,

$$F\Big((a_k)_{k \in \mathbb{N}}\Big) \in \Big\{ (i_k)_{k \in \mathbb{N}} \subseteq \mathbb{N} \ \Big|\ i_k \text{ is increasing in }k \land a_{i_k} \text{ is non-decreasing in }k \Big\}.$$

That being set, let's make your infinite set $S \subseteq \mathbb{N}^n$ of tuples into an infinite sequence $S^0$ of tuples and then let $S^i$ be a subsequence of $S^{i-1}$ such that its $i$-th coordinate is non-decreasing

$$S^{i}_k = S^{i-1}_{F(\pi_i \circ S^{i-1})_k}.$$

Finally, any two tuples of $S^n$ would give you what you want.

I hope this helps $\ddot\smile$