Triangle inequality…

Note that:

$$– \left| x \right| \leq x \leq \left| x \right|$$

$$– \left| y \right| \leq y \leq \left| y \right|$$

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

Use the fact that $$\left| b \right| \leq a \Leftrightarrow – a \leq b \leq a$$ (with $$b$$ replaced by $$x+y$$ and $$a$$ by $$\left| x \right| + \left| y \right|$$), we have

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

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 $$\left| {z – y} \right| = \left| {y – z} \right|$$: