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Q&A 1 picture proof for the triangle inequality and the reverse triangle inequality

0 answers · posted 1y ago by Este‭ · edited 1y ago by trichoplax‭

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#7: Post edited by $user avatar$ trichoplax‭ · 2023-08-22T14:17:23Z (over 1 year ago)
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1 picture proof for $|x-y| ≤ |x|+|y| \& \color{green}{|x|-|y|} ≤ |x-y| \& {\color{magenta}{|}}{\color{green}{|x|-|y|}}{\color{magenta}{|}} ≤ |x-y|$?

1 picture proof for the triangle inequality and the reverse triangle inequality

Michael Spivak's *Calculus* (2008 4 edn), Exercise 12, p 16.   
(vi) is the Reverse Triangle Inequality. 

>(iv) $|x-y| ≤ |x| + |y|$. (Give a very short proof.)    
(v) $\color{limegreen}{|x|-|y|} ≤ |x-y|$. (A very short proof is possible, if you write things in the right way).   
(vi)  ${\color{magenta}{|}}({\color{limegreen}{|x|-|y|}}){\color{magenta}{|}} ≤ |x  - y|$ (Why does this follow immediately from (v)?)

How can I simultaneously pictorialize (iv) and (vi), in [this picture proof](https://math.stackexchange.com/a/3768046) for (v) above? To wit, how can I pictorialize (iv), (v), (vi) together in the same sole picture proof?

![](https://i.stack.imgur.com/gOjSE.jpg)

I fancy  "killing" 3 inequalities with 1 picture proof. I'm NOT seeking 3 picture proofs! ["You can change < by ≤, BUT NOT ≤ by <"](https://math.stackexchange.com/a/505354).

#6: History hidden by $user avatar$ Este‭ · 2023-08-22T08:04:06Z (over 1 year ago)
Detailed history before this event is hidden because of a redaction.

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1 picture proof for $|x-y| ≤ |x|+|y| \& \color{green}{|x|-|y|} ≤ |x-y| \&  {\color{magenta}{|}}{\color{green}{|x|-|y|}}{\color{magenta}{|}} ≤ |x-y|$?

Michael Spivak's *Calculus* (2008 4 edn), Exercise 12, p 16.   
(vi) is the Reverse Triangle Inequality. 

>(iv) $|x-y| ≤ |x| + |y|$. (Give a very short proof.)    
(v) $\color{limegreen}{|x|-|y|} ≤ |x-y|$. (A very short proof is possible, if you write things in the right way).   
(vi)  ${\color{magenta}{|}}({\color{limegreen}{|x|-|y|}}){\color{magenta}{|}} ≤ |x  - y|$ (Why does this follow immediately from (v)?)

How can I simultaneously pictorialize (iv) and (vi), in [this picture proof](https://math.stackexchange.com/a/3768046) for (v) above? To wit, how can I pictorialize (iv), (v), (vi) together in the same sole picture proof?

![](https://i.stack.imgur.com/gOjSE.jpg)

I fancy  "killing" 3 inequalities with 1 picture proof. I'm NOT seeking 3 picture proofs! ["You can change < by ≤, BUT NOT ≤ by <"](https://math.stackexchange.com/a/505354).

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#5: Post edited by $user avatar$ Este‭ · 2023-08-22T08:04:06Z (over 1 year ago)

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