Proof using Triangle inequality

**eXist** · September 29th 2009, 02:07 PM

I have the problem:

Prove:
$||x| - |y|| \le |x - y|$

And I know I'm suppose to use the triangle inequality and the fact:
$x = x + y - y$ and $y = y + x - x$

I tried starting with:

$|x - y| =>$
$|(x - y + y) - (y - x + x)|$

But I don't know how to break this up using the triangle inequality:
$|x + y| \le |x| + |y|$

**Plato** · September 29th 2009, 02:22 PM

$\left| x \right| = \left| {x - y + y} \right| \leqslant \left| {x - y} \right| + \left| { y} \right|\; \Rightarrow \;\left| x \right| - \left| y \right| \leqslant \left| {x - y} \right|$

Likewise: $\left| y \right| = \left| {y - x + x} \right| \leqslant \left| {y - x} \right| + \left| x \right|\; \Rightarrow \;\left| y \right| - \left| x \right| \leqslant \left| {y - x} \right|$

Recall that $|x-y|=|y-x|$

So putting the first two together we get.
$- \left| {x - y} \right| \leqslant \left| x \right| - \left| y \right| \leqslant \left| {x - y} \right|\; \Rightarrow \;\left| {\left| x \right| - \left| y \right|} \right| \leqslant \left| {x - y} \right|$ .

**eXist** · September 29th 2009, 03:03 PM

I'm so upset with myself. I honestly tried that but never thought of moving $|y|$ to the LHS. :/........

Thanks so much

. You're my hero.

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