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Question Number 127400 by mohammad17 last updated on 29/Dec/20
provethat:∣∣z1∣−∣z2∣∣⩽∣z1+z2∣
Commented by mohammad17 last updated on 29/Dec/20
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Commented by liberty last updated on 29/Dec/20
observasitheinequality∣z1+z2∣⩽∣z1∣+∣z2∣isknownastriangleinequality.Nowfromtheidentityz1=z1+z2+(−z2)gives∣z1∣=∣z1+z2+(−z2)∣⩽∣z1+z2∣+∣−z2∣since∣z2∣=∣−z2∣then∣z1∣−∣z2∣⩽∣z1+z2∣...(i)butbecausez1+z2=z2+z1,(i)canbewritteninthealternativeform∣z1+z2∣=∣z2+z1∣⩾∣z2∣−∣z1∣=−(∣z1∣−∣z2∣)or∣z2∣−∣z1∣=∣∣z1∣−∣z2∣∣...(ii)replacingz2by−z2weget∣z1−z2∣⩾∣∣z1∣−∣−z2∣∣or∣z1−z2∣⩾∣∣z1∣−∣z2∣∣
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