Showing $\left | d\left ( u,v \right )-d\left ( w,x \right ) \right |\leq d\left ( u,w \right )+d\left ( v,x \right )$

Question

Lemma:

Let $u,v,w,x \in X$. Then, $\left | d\left ( u,v \right )-d\left ( w,x \right ) \right |\leq d\left ( u,w \right )+d\left ( v,x \right )$

By the triangle inequality:

$d\left ( u,v \right )-d\left ( w,x \right )\leq d\left ( u,w \right )+d\left ( x,v \right )$ and $d\left ( w,x \right )-d\left ( u,v \right )\leq d\left ( w,u \right )+d\left ( v,x \right )$

Any help is appreciated. Thanks in advance.

Answer 1

Using the triangle inequality (and the symmetrie of your metric $d$) is already enought since $$|x|\leq c \qquad \Longleftrightarrow \qquad -x \leq c \ \text{and} \ x\leq c.$$