Simple Euclidean Norm Inequality

Question

I feel rather silly for having to ask this question in specific and am by no means looking for a flat out step by step answer. I understand the definition for the euclidean norm in an n-dimensional space (as defined here). I can't figure out how to apply it however to even a simple problem like this one:

If $\| x - z \| \lt 2$ and $\| y - z \| \lt 3$, prove $\| x - y \| \lt 5$.

Sorry in advance for lack of formatting, I'm new to math exchange and there's really nothing complicated to format. Again, I am not looking for a straight answer. My proof breaks down after I add the two assumptions and attempt to square both sides. Hopefully someone can point out the simple first step here. Thanks.

Answer 1

Apply the triangle inequality $\|a+b\| \leq \|a\| + \|b\|$, with $a=x-z$ and $b=z-y$.

$$ \|x-y\| = \| x -z + z - y \| = \| (x -z) + (z - y) \| \leq \| (x -z) \| + \|(z - y) \| < 2+3=5 \\ \implies \|x-y\| < 5.$$

Answer 2

First observe that $\ \|x-z\| < 2$ and $\ \|y-z\| < 3 \Rightarrow \ \|x-z\| + \ \|y-z\| < 5$.

So, using triangle inequality, we have $\ \|x-y\| = \ \|x-z+z-y\| \leq \ \|x-z\| + \ \|y-z\| < 5 $.