Triangle inequality…

Note that:

$- \left| x \right| \leq x \leq \left| x \right|$

$- \left| y \right| \leq y \leq \left| y \right|$

After adding,

$- \left( {\left| x \right| + \left| y \right|} \right) \leq x + y \leq \left( {\left| x \right| + \left| y \right|} \right)$

$\left| {x + y} \right| \leq \left| x \right| + \left| y \right|$

Now, note that

$\left| {x - y} \right| = \left| {\left( {x - z} \right) + \left( {z - y} \right)} \right|$

Now we use tre triangle inequality and the fact that $\left| {z - y} \right| = \left| {y - z} \right|$ :