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

Prove the following properties of real numbers.

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