Proving inequality with vectors

Question

I'm struggling with the proof of

$\|x-z\| \leq \|x-y\| + \|y-z\|$ for $x, y, z \in \mathbb{R}^{n}$

I tried using the definitions of the norm, also tried squaring and simplifying both sides but I don't get the desired result. Could someone give me a hint?

Answer 1

By triangle inequality for the norm in $\mathbb{R}^n$, we have:

$$\|x-z\|=\|x-y+y-z\| \leq \|x-y\| + \|y-z\|$$