Real numbers/Modulus/Properties/Fact/Proof/Exercise

Prove the following properties for the absolute value function

$\mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto \vert {x}\vert ,$

(here let ${}x,y$ be arbitrary real numbers).

1. ${}\vert {x}\vert \geq 0$.
2. ${}\vert {x}\vert =0$ if and only if ${}x=0$.
3. ${}\vert {x}\vert =\vert {y}\vert$ if and only if ${}x=y$ or ${}x=-y$.
4. ${}\vert {y-x}\vert =\vert {x-y}\vert$.
5. ${}\vert {xy}\vert =\vert {x}\vert \vert {y}\vert$.
6. For ${}x\neq 0$ we have ${}\vert {x^{-1}}\vert =\vert {x}\vert ^{-1}$.
7. We have ${}\vert {x+y}\vert \leq \vert {x}\vert +\vert {y}\vert$ (triangle inequality for modulus).
8. ${}\vert {x+y}\vert \geq \vert {x}\vert -\vert {y}\vert$.

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