I am working on a hw exercise and would like to check if I understand it fully. I don't trust myself.
Problem: Let $n^2 + 1$ points be given in $\mathbb{R}^2$. Prove that there is a sequence of $n + 1$ points $(x_1, y_1), \ldots, (x_{n+1}, y_{n+1})$ for which $x_1 \leq x_2 \leq \cdots \leq x_{n+1}$ and $y_1 \geq y_2 \geq \cdots \geq y_{n+1}$ or a sequence of $n + 1$ points for which $x_1 \leq x_2 \leq \cdots \leq x_{n+1}$ and $y_1 \leq y_2 \leq \cdots \leq y_{n+1}$.
Proof: Define a partial ordering $\preceq$ over $\mathbb{R^2}$ by $(x_1, y_1) \preceq (x_2, y_2) \implies x_1 < x_2$ or $x_1 = x_2 \text{ and } y_1 \leq y_2$. Apply Dilworths Lemma, which says that given a poset of size $k \geq rs + 1$ there will be a chain of length $s + 1$ or anti-chain of length $r + 1$. Let $s = r = n$ and we have $n^2 + 1$ elements required along with the existence of a chain or anti-chain of length $n + 1$.
Am I missing any steps?