# Reelle Zahlen/Anordnungseigenschaften/Folgerungen/Fakt/Beweis/Aufgabe/en

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Prove the following properties of real numbers.

1. ${\displaystyle {}1>0}$.
2. From ${\displaystyle {}a\geq b}$ and ${\displaystyle {}c\geq 0}$ it follows ${\displaystyle {}ac\geq bc}$.
3. From ${\displaystyle {}a\geq b}$ and ${\displaystyle {}c\leq 0}$ it follows ${\displaystyle {}ac\leq bc}$.
4. ${\displaystyle {}a^{2}\geq 0}$ holds.
5. From ${\displaystyle {}a\geq b\geq 0}$ it follows ${\displaystyle {}a^{n}\geq b^{n}}$ for all ${\displaystyle {}n\in \mathbb {N} }$.
6. From ${\displaystyle {}a\geq 1}$ it follows ${\displaystyle {}a^{n}\geq a^{m}}$ für ganze Zahlen ${\displaystyle {}n\geq m}$.
7. From ${\displaystyle {}a>0}$ it follows ${\displaystyle {}{\frac {1}{a}}>0}$.
8. From ${\displaystyle {}a>b>0}$ it follows ${\displaystyle {}{\frac {1}{a}}<{\frac {1}{b}}}$.