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Triangle inequality…

Note that:

\( – \left| x \right| \leq x \leq \left| x \right|\)

\( – \left| y \right| \leq y \leq \left| y \right|\)

After adding,

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

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 \(\left| {z – y} \right| = \left| {y – z} \right|\):

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