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?
2026-04-02 03:17:10.1775099830
How to obtain $||x|-|y||\le|x-y|$ from $|x|-|y|\le |x-y|$?
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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.