Triangle Inequation proof

Question

I have to show for all x,y ∈ ℝ that ||x| - |y| ≤ |x-y|

I know that |x| = |x - y + y| ≤ |x - y|+|y|

Does it suffice to put in numbers now or how would I proof that?

From what you know. it is enough to transform your inequality slightly. You want $|x-y|$ alone on the right side — Hagen von Eitzen, Nov 23 '20 at 20:52
Does this answer your question? Proof of triangle inequality — collins - supports Monica, Nov 23 '20 at 20:53
Well it seems that you finished the exercice, just put $|y|$ on the left side of the equation. — yrual, Nov 23 '20 at 20:54

score 0 · Answer 1 · answered Nov 23 '20 at 20:55

0

We use the fact that

$$a\le b \; and \; -a\le b \;\implies$$ $$-b\le a\le b \implies |a|\le |b|$$

So, as you wrote $$|x|=|x-y+y|\le |x-y|+|y| \implies $$

$$|x|-|y|\le |x-y|$$ and by symetry, $$|y|-|x|\le |x-y|$$

take $ a=|x|-|y|\; and \; b=|x-y|$.

answered Nov 23 '20 at 20:55

hamam_Abdallah

score 0 · Answer 2 · answered Nov 23 '20 at 21:41

You have $|x| = |x - y + y| ≤ |x - y|+|y|$ so $|x|-|y| \leq |x-y|$

Also $|y| = |y-x+x| \leq |y-x|+|x|$ and by subtracting $|y-x|$ and $|y|$ from both sides one gets $-|x-y|=-|y-x| \leq |x|-|y|$

Since $-|x-y| \leq |x|-|y| \leq |x-y| $,$ ||x|-|y|| \leq |x-y|$.

2 Answers2