Q:

Showing a function is a metric on $\mathbb{R}^{2}$

Question:
For $x=\left ( x_{1},x_{2} \right ),y=\left ( y_{1},y_{2} \right ) \in \mathbb{R}^{2}$
Define $d\left ( x,y \right )=\max\left \{ \left | x_{1}-y_{1} \right |,\left | x_{2}-y_{2} \right | \right \}$.
Show that $d$ is a metric on $\mathbb{R}^{2}$.

So, I've shown all the axioms except the triangle inequality which apparently is giving me a bit of problem.
Any help is appreciated.
Thanks in advance.

A:

$|x_1 - y_1| \leq |x_1 - z_1| + |z_1 - y_1| \leq d(x,z) + d(z,y),$ and similarly for $|x_2 - y_2|$. Therefore
$d(x,y) = max\{|x_1 - y_1|,|x_2 - y_2|  \} \leq d(x,z) + d(y,z)$

