Some Strange Minimum and proof

Question

I read following on Norm chapter in one book.

$$\begin{align}\left|\left\|x-y\right\|-\left\|w-z\right\|\right| \leq & \min \{\|x-w\| + \|y-z\| , \|x-z\| + \|y-w\|\}\\\text{ or, }&\min \{\|x-z\| + \|y-w\| , \|x-y\|+\|w-z\| \}\end{align}$$

I need some proof or hint's how this is true...

Answer 1

It is enough to show that: $$\left\|x-y\right\|-\left\|w-z\right\|\leq \|x-w\| + \|y-z\| $$ $$\left\|w-z\right\|-\left\|x-y\right\|\leq \|x-w\| + \|y-z\| $$ $$\left\|x-y\right\|-\left\|w-z\right\|\leq \|x-z\| + \|y-w\| $$ $$\left\|w-z\right\|-\left\|x-y\right\|\leq \|x-z\| + \|y-w\|$$ and so on. But this follows directly from triangle inequality. For example, the first inequality: $$\left\|x-y\right\|\leq \left\|(w-z)+(x-w)+(z-y)\right\|\leq \left\|w-z\right\|+ \|x-w\| + \|y-z\| $$