triangle inequality

**PersaGell** · June 7th 2007, 02:00 PM

How do I prove the following identity?

| |x| -|y| | <= |x - y|

Thank you

**ThePerfectHacker** · June 7th 2007, 02:10 PM

Originally Posted by PersaGell

How do I prove the following identity?

| |x| -|y| | <= |x - y|

Thank you

$|xy|\geq xy$

$-2|x||y|\leq -2xy$

$x^2-2|x||y|+y^2 \leq x^2 - 2xy + y^2$

$0 \leq (|x|-|y|)^2 \leq (x-y)^2$

$\sqrt{(|x|-|y|)^2} \leq \sqrt{(x-y)^2}$

$||x|-|y||\leq |x-y|$

**Plato** · June 7th 2007, 02:25 PM

Here is a second way.

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