I cannot figure out how to prove $||x| - |y|| \leq |x - y|$

Question

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.

do you know the triangle inequality $|u|+|y|\le|u+y|$? take $u=x-y$ — J. W. Tanner, Sep 05 '19 at 18:42
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 — J. W. Tanner, Sep 05 '19 at 21:12

J. W. Tanner · Accepted Answer · 2019-09-05T18:58:24.320

0

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

edited Sep 05 '19 at 18:58

answered Sep 05 '19 at 18:46

J. W. Tanner

1 Answers1