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I had a quiz today in my Real Analysis course asking to prove:

for two vectors $x, y \in \Bbb{R^n}$, show that $||x| - |y|| \leq |x - y|$. When does equality hold?

... I cannot figure it out for the life of me. It's really frustrating. Any help would be appreciated.

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    $\begingroup$ that's the reverse triangle inequality $\endgroup$ Commented Sep 5, 2019 at 18:40
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    $\begingroup$ $|x|=|(x-y)+y|\leq|x-y|+|y|$ and so on ... $\endgroup$
    – rtybase
    Commented Sep 5, 2019 at 18:41
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    $\begingroup$ do you know the triangle inequality $|u|+|y|\le|u+y|$? take $u=x-y$ $\endgroup$ Commented Sep 5, 2019 at 18:42
  • $\begingroup$ hint, if the vectors are colinear the equality is true $\endgroup$
    – dmtri
    Commented Sep 5, 2019 at 18:43
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    $\begingroup$ the triangle inequality involves the absolute values of three quantities, one of which is the sum of the other two; this inequality involves the absolute values of three quantities, one of which is the difference of the other two; if you define a new variable to be the sum of the other two, then one of the others becomes a difference $\endgroup$ Commented Sep 5, 2019 at 21:12

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The reverse triangle inequality can be proved from the triangle inequality

$|u+y|\le|u|+|y|$ or $|u+x|\le|u|+|x|$

by taking $u=x-y$ and $u=y-x$:

$|x|\le|x-y|+|y|$ so $|x|-|y|\le|x-y|$ and $|y|-|x|\le |y-x|$.

Therefore, $||x|-|y||\le|y-x|=|x-y|$.

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  • $\begingroup$ This is just the top answer in the linked dupe. $\endgroup$
    – Randall
    Commented Sep 5, 2019 at 18:57

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