Prove that $|p_1 - q_1| \leq p_2 - q_2$ relation is transitive for all points in $\mathbb{R}^2$

Question

Given two points $p,q \in \mathbb{R}^2$, we say $p \preccurlyeq q$ if $|p_1 - q_1| \leq p_2 - q_2$.
How do I show that this relation is transitive?
By definition I have to show that if xRy and yRz then xRz $\forall x,y,z \in \mathbb{R}^2$.

Answer 1

Suppose $p \preccurlyeq q$ and $q \preccurlyeq r$, we want to show that $p \preccurlyeq r$. Using the triangle inequality, we have:

$|p_1 - r_1| = |p_1 - q_1 + q_1 - r_1| \leq |p_1 - q_1| + |q_1 - r_1| \leq p_2 - q_2 + q_2 - r_2 = p_2 - r_2$

Therefore, one has $p \preccurlyeq r$.