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Real numbers/Ordering/Properties/Fact/Proof/Exercise
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Real numbers
Prove the following properties of real numbers.
1
≥
0
{\displaystyle {}1\geq 0}
.
From
a
≥
b
{\displaystyle {}a\geq b}
and
c
≥
0
{\displaystyle {}c\geq 0}
follows
a
c
≥
b
c
{\displaystyle {}ac\geq bc}
.
From
a
≥
b
{\displaystyle {}a\geq b}
and
c
≤
0
{\displaystyle {}c\leq 0}
follows
a
c
≤
b
c
{\displaystyle {}ac\leq bc}
.
a
2
≥
0
{\displaystyle {}a^{2}\geq 0}
holds.
a
≥
b
≥
0
{\displaystyle {}a\geq b\geq 0}
implies
a
n
≥
b
n
{\displaystyle {}a^{n}\geq b^{n}}
for all
n
∈
N
{\displaystyle {}n\in \mathbb {N} }
.
From
a
≥
1
{\displaystyle {}a\geq 1}
follows
a
n
≥
a
m
{\displaystyle {}a^{n}\geq a^{m}}
for integers
n
≥
m
{\displaystyle {}n\geq m}
.
From
a
>
0
{\displaystyle {}a>0}
follows
1
a
>
0
{\displaystyle {}{\frac {1}{a}}>0}
.
From
a
>
b
>
0
{\displaystyle {}a>b>0}
follows
1
a
>
1
b
{\displaystyle {}{\frac {1}{a}}>{\frac {1}{b}}}
.
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