(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

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: 2 Proofs: Does this work?

**Physics Forums | Science Articles, Homework Help, Discussion**