Proving if something is a half norm

Question

There is a function given: $$d(x) = \sqrt{|x_2-x_1|^2+|x_3-x_2|^2}, x \in \mathbb{R^3}.$$ I am to proof if $d(x+y) \le d(x) + d(y)$ where $x, y \in \mathbb{R^3}$.
I've tried to prove the statement above in several ways. Unfortunately I've failed. I would appreciate any hints or tips.

Answer 1

$$ d(x) = \left\| \pmatrix{x_1\\x_2} - \pmatrix{x_2\\x_3} \right\|_2 $$ where $\|\cdot\|_2$ is the Euclidean norm on $\mathbb R^2$

which satisfies a triangle inequality