On p. 12, Michael Spivak's Calculus (2008 4 edn) proved $|x + y| ≤ \color{darkgoldenrod}{|x| + |y|}$ (Triangle Inequality). Ibid, exercise 12, p. 16.
(iv) ${\color{red}{|x-y|}} ≤ \color{goldenrod}{|x| + |y|}$. (Give a very short proof.)
(v) ${\color{limegreen}{|x|-|y|}} ≤ {\color{red}{|x-y|}}$. (A very short proof is possible, if you write things in the right way).
(vi) $\left|{\color{limegreen}{|x|-|y|}}\right| ≤ |x - y|$ (Why does this follow immediately from (v)?)
[This (vi) is Reverse Triangle Inequality, but I deleted Spivak's superfluous set of round brackets.]
How can one and same picture prove all 4 inequalities above? Please contrast $\vec{x}, \vec{y}$ so that visibly, $|\vec{x} - \vec{y}| < |\vec{x} + \vec{y}|$. You can use my improvement of this.
