Proving that $d(x,y) =\max_i{\lvert x_i - y_i \rvert }$ is a distance in $\mathbb{R}^2$

Question

I was asked to prove that $d(x,y) =\max_i{\lvert x_i - y_i \rvert }$ is a distance function in $\mathbb{R}^2$. I've got myself stuck with proving the triangle inequality.

Can someone give me an hint about how to proceed?

Answer 1

$$ |x_i - z_i|\leq |x_i - y_i| + |y_i - z_i| \quad\forall i \qquad\implies $$

$$ \max_i|x_i - z_i|\leq \max_i\left(|x_i - y_i| + |y_i - z_i|\right) \leq \max_i|x_i - y_i| + \max_i |y_i - z_i| $$