1 picture proof for the triangle inequality and the reverse triangle inequality
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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 for (v) above? To wit, how can I pictorialize (iv), (v), (vi) together in the same sole picture proof?
I fancy "killing" 3 inequalities with 1 picture proof. I'm NOT seeking 3 picture proofs! "You can change < by ≤, BUT NOT ≤ by <".
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