(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Prove the following:

(i) ##|x|-|y| \le |x-y|##

and

(ii) ##|(|x|-|y|)| \le |x-y|\qquad## (Why does this immediately follow from (i) ?)

2. Relevant equations

##|z| = \sqrt{z^2}##

3. The attempt at a solution

(i) ##(|x|-|y|)^2 = |x|^2 - 2|x||y| + |y|^2 = x^2 - 2|x||y| + y^2 \le x^2 - 2xy + y^2= (x-y)^2 \implies \boxed{|x|-|y| \le |x-y|.}##

(ii) For this part, I looked at the question "Why does this immediately follow from (i)" for inspiration and saw that if I could show that ##|(|x|-|y|)| \le |x-y|## then the proof is complete by transitivity.

Is it as simple as:

##|(|x|-|y|)| = \sqrt{(|(|x|-|y|)|)^2} = \sqrt{(|x|-|y|)^2} = |x|-|y|?##

I think that it is, but it is getting late

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# 2 Proofs: Does this work?

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