Max function is a metric?

Question

I was wondering if the max function is a metric or if, in particular, $\max(|x + y|, 1)$ is equal or less than $\max(|x|, 1) + \max(|y|, 1)$ with $x$ and $y$ belonging to $R$.

Answer 1

Proof by cases:

Case 1: If $|x+y|\geq 1$, then:

$\max(|x+y|, 1)=|x+y|\leq|x|+|y|\leq\max(|x|, 1)+\max(|y|, 1)$

Case 2: $|x+y|<1$, then:

$\max(|x+y|, 1)=1<1+1\leq\max(|x|, 1)+\max(|y|, 1)$