Having the following inequality: $$|x|-|y|\le |x-y|$$ does it imply that $||x|-|y||\le|x-y|$ if it does (i think it does) how to prove it?
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1Consider three cases: $\lvert x\rvert <\lvert y\rvert$, $\lvert x\rvert = \lvert y\rvert$, and $\lvert x\rvert > \lvert y\rvert$. – Jon Feb 14 '15 at 17:24
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mea culpa, i am sorry – kurkowski Feb 14 '15 at 17:28
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@kurkowski - Not your fault. It took me a while to find. – Clarinetist Feb 14 '15 at 17:40
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The triangle inequality is $$|a+b|\le|a|+|b|.$$ Let $a=x-y$ and $b=y$. Then $a+b=x$. Our above inequality becomes $$|x|\le|x-y|+|y|.$$ Now subtract $|y|$ on both sides to arrive at your result.
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