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How can I prove using the triangular inequality that: $$|a+b| \geq |a| - |b|$$

I already proved it by considering all 8 possible scenarios (like a>b and b=0 ... etc) However I couldn’t manage to find the way to prove it only by using the triangular inequality.

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    $\begingroup$ This is the "reverse triangle inequality". See here or here for example. $\endgroup$ Commented Mar 23, 2019 at 0:21

2 Answers 2

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Hint: the inequality you wish to prove is equivalent to $$|a + b| + |-b| \ge |a|.$$

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By triangular inequality we have: $$|a|=|a+b +(-b)|\leq |a+b|+|b|.$$ Thus, $$|a+b| \geq |a| - |b|.$$

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