# I cannot figure out how to prove $||x| - |y|| \leq |x - y|$ [duplicate]

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.

• that's the reverse triangle inequality Commented Sep 5, 2019 at 18:40
• $|x|=|(x-y)+y|\leq|x-y|+|y|$ and so on ... Commented Sep 5, 2019 at 18:41
• do you know the triangle inequality $|u|+|y|\le|u+y|$? take $u=x-y$ Commented Sep 5, 2019 at 18:42
• hint, if the vectors are colinear the equality is true Commented Sep 5, 2019 at 18:43
• 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 Commented Sep 5, 2019 at 21:12

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