Proving this distance to be a metric?

First, note that $|x-y|$ is the usual Euclidean distance between two points $x,y\in\mathbb{R}$. Then $\lvert x-y\rvert\le \lvert x-z\rvert+\lvert z-y\rvert$, which is the standard triangle inequality. Using this, $$ \sqrt{\lvert x-y\rvert}\le \sqrt{\lvert x-z\rvert+\lvert z-y\rvert}\le \sqrt{\lvert x-z\rvert}+\sqrt{\lvert z-y\rvert}$$ by the relation you stated. This is the triangle inequality you desire.

In general you can prove that if $(X,d)$ is a metric space, and $f:\mathbb R_{>0}\to\mathbb R_{>0}$ satisfies the conditions

• $f(x)=0$ if and only if $x=0$.
• $f$ is (weakly) concave.

then $f\circ d$ is a metric.

The triangle inequality follows from the triangle inequality for $d$ because $f(p)+f(q)\geq f(p+q)$.

In your case, take $f(x)=\sqrt x$.

By squaring we have $$\sqrt {|x-y|}\leq \sqrt {|x-z|}+\sqrt {|z-x|}\iff$$ $$ |x-y|\leq |x-z|+|z-y|+2\sqrt {|x-z|} \sqrt {|z-y|}$$ which is immediate from $|x-y|=|(x-z)+(z-y)|\leq |x-z|+|z-x|.$