Question Number 65743 by aliesam last updated on 03/Aug/19
Commented by ~ À ® @ 237 ~ last updated on 04/Aug/19
![On a 1=∣1∣=∣1+(y−z)−(y−z)∣ =∣1+y−z −[(y−x)+(x−z)]∣ = ∣ (1+y−z)+(x−y)+(−(x−z))∣ quand tu appliques la sous additivite 1< ∣1+y−z∣+∣x−y∣+∣−(x−z)∣ cqfd](Q65820.png)
$$\:\: \\ $$$${On}\:{a} \\ $$$$\mathrm{1}=\mid\mathrm{1}\mid=\mid\mathrm{1}+\left({y}−{z}\right)−\left({y}−{z}\right)\mid \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:=\mid\mathrm{1}+{y}−{z}\:−\left[\left({y}−{x}\right)+\left({x}−{z}\right)\right]\mid \\ $$$$\:\:\:\:\:\:\:\:\:\:\:=\:\mid\:\left(\mathrm{1}+{y}−{z}\right)+\left({x}−{y}\right)+\left(−\left({x}−{z}\right)\right)\mid \\ $$$${quand}\:{tu}\:{appliques}\:{la}\:{sous}\:{additivite}\: \\ $$$$\mathrm{1}<\:\mid\mathrm{1}+{y}−{z}\mid+\mid{x}−{y}\mid+\mid−\left({x}−{z}\right)\mid\:\:\:\:{cqfd} \\ $$