Reelle Zahlen/Betragseigenschaften/Fakt/Beweis/Aufgabe/en

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Prove the following properties for the absolute value function

${\displaystyle \mathbb {R} \longrightarrow \mathbb {R} ,\,x\longmapsto \vert {x}\vert ,}$
(here let ${\displaystyle {}x,y}$ be arbitrary real numbers).
1. ${\displaystyle {}\vert {x}\vert \geq 0}$.
2. ${\displaystyle {}\vert {x}\vert =0}$ if and only if ${\displaystyle {}x=0}$.
3. ${\displaystyle {}\vert {x}\vert =\vert {y}\vert }$ if and only if ${\displaystyle {}x=y}$ or ${\displaystyle {}x=-y}$.
4. ${\displaystyle {}\vert {y-x}\vert =\vert {x-y}\vert }$.
5. ${\displaystyle {}\vert {xy}\vert =\vert {x}\vert \vert {y}\vert }$.
6. For ${\displaystyle {}x\neq 0}$ we have ${\displaystyle {}\vert {x^{-1}}\vert =\vert {x}\vert ^{-1}}$.
7. We have ${\displaystyle {}\vert {x+y}\vert \leq \vert {x}\vert +\vert {y}\vert }$ (Triangle inequality for the absolute value).