How to proof that $\left|x - y\right| + \left|y - z\right| \geq \left|x - z\right|$, where $x, y, z \in \mathbb{R}$?

Asked Dec 01 '21 at 15:21

Active Dec 01 '21 at 15:21

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I am not a professional mathematician and I recently started studying math as a hobby. Intuitively this result is true. My intuition is the distance from x to y plus the distance from y to z cannot be strictly less than the distance from x to z. However, I have no idea how to prove this proposition.

$$\left|x - y\right| + \left|y - z\right| \geq \left|x - z\right|, \forall x, y, z \in \mathbb{R}$$

asked Dec 01 '21 at 15:21

Raphael Augusto

Ever heard of the triangle inequality? – JohnK Dec 01 '21 at 15:23
hint: $|x-z|=|x-y+y-z|$ – Masacroso Dec 01 '21 at 15:24
Simply use the definition of the absolute value. – principal-ideal-domain Dec 01 '21 at 15:24
@JohnK no, that's what I was looking for. Thanks!!! – Raphael Augusto Dec 01 '21 at 15:25

How to proof that $\left|x - y\right| + \left|y - z\right| \geq \left|x - z\right|$, where $x, y, z \in \mathbb{R}$?

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