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Question Number 193080 by Mastermind last updated on 03/Jun/23

Show that for a,b∈R, ∣a−b∣≥∣a∣−∣b∣ (Hint: write a=(a−b)+b

Showthatfora,b∈R,∣a−b∣⩾∣a∣−∣b∣(Hint:writea=(a−b)+b

Answered by Subhi last updated on 03/Jun/23

∣a∣−∣b∣ = ∣(a−b)+b)∣−∣b∣ note that: ∣a+b∣≤∣a∣+∣b∣ ∣(a−b)+b∣−∣b∣≤∣(a−b)∣+∣b∣−∣b∣=∣a−b∣ ∴ ∣a−b∣≥∣a∣−∣b∣

∣a∣−∣b∣=∣(a−b)+b)∣−∣b∣notethat:∣a+b∣⩽∣a∣+∣b∣∣(a−b)+b∣−∣b∣⩽∣(a−b)∣+∣b∣−∣b∣=∣a−b∣∴∣a−b∣⩾∣a∣−∣b∣

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