How do I show that $ d((x_1, x_2), (y_1, y_2)) = |y_1 - x_1| + |y_2 - x_2|$ is a metric? I've managed to show the first two axioms that are required for a set to be a metric space. But can't seem to get past showing that for $ z_1, z_2 \in \mathbf R$, $$ d((x_1,x_2),(y_1,y_2)) + d((y_1, y_2),(z_1, z_2)) = |y_1 - x_1| + |y_2 - x_2| + |z_1 - y_1| + |z_2 - y_2|$$ but I don't know how to show that this sum is larger than $d((x_1,x_2),(z_1,z_2))$
Sorry if this question is a duplicate.

Hint:
$$d((x_1,y_1),(x_2,y_2)) =|x_1-z_1+z_1-y_1|+|x_2-z_2+z_2-y_2|. $$
