How do I show that $ ||x| - |y||\leq|x - y| $

Question

How do I show that $$ ||x| - |y||\leq|x - y| $$

I can see obviously that $ ||x| - |y||\le |x|-|y|.$ I just can't figure this one out.

Something's wrong: your claim is false... For example, $|1+|(-1)||$ is not less-than-or-equal $|1+(-1)|=|0|=0$ — paul garrett, Jul 20 '18 at 01:05
Lookup the reverse triangle inequality. Btw, both obviously claims are false. — dxiv, Jul 20 '18 at 01:06
In response to your edit, the inequality $|x-y| \leq |x|-|y|$ now fails, for instance with $x=1$ and $y=-1$. — B. Mehta, Jul 20 '18 at 01:06

score 3 · Accepted Answer · answered Jul 20 '18 at 01:09

3

Sketch: Notice first that if you can show $|x| - |y| \leq |x-y|$ and $|y| - |x| \leq |x-y|$, then you're done. Also observe that these two inequalities are ultimately the same, up to relabelling so it suffices to show $|x| - |y| \leq |x-y|$. But now, rearrange this to $|x| \leq |y| + |x-y|$ which should hopefully look familiar to you.

answered Jul 20 '18 at 01:09

B. Mehta

12,774

Possible duplicate of Reverse Triangle Inequality Proof – Math Lover Jul 20 '18 at 01:43

How do I show that $ ||x| - |y||\leq|x - y| $

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