If $X=\mathbb{R}$ and $d\colon\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ given by $d(x,y)={\sqrt{|x-y|}}$, I am able to show that this is indeed a metric.
But what I don't understand is how this metric doesn't come from a norm, like how do I even define a norm on this metric? Thanks
We have that
$$d(x,0)=\sqrt {|x|}.$$ Thus
$$d(\lambda x,0)=\sqrt{|\lambda|}\sqrt {|x|}\ne |\lambda| d(x,0)$$ if $|\lambda|\ne \pm 1$ and $x\ne 0.$ That is, $d$ is a metric but not comes from a norm.