prove-that-a-a-a-a-is-a-real-number-

Question Number 72900 by aliesam last updated on 04/Nov/19

p r o v e t h a t

- ∣ a ∣⩽ a ⩽∣ a ∣

a i s a r e a l n u m b e r

Answered by MJS last updated on 04/Nov/19

\forall r \in R : ∣ r ∣:= {\begin{cases} r; r ⩾ 0 \\ - r; r < 0 \end{cases}

\Rightarrow {\begin{cases} a ⩾ 0 : \pm ∣ a ∣= \pm a \Rightarrow - ∣ a ∣⩽ a =∣ a ∣ \\ a < 0 : \pm ∣ a ∣= \mp a \Rightarrow - ∣ a ∣= a <∣ a ∣ \end{cases}

\Rightarrow \forall a \in R : - ∣ a ∣⩽ a ⩽∣ a ∣

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