$\forall x,y \in \mathbb R$ if distance $d(x,y) = \sqrt { |(x - y)|}$,
How can I prove this distance to be metric? I am stuck at triangular equality.
i.e. $ \sqrt{ |x - y| } \leq \sqrt{ | x - z|} + \sqrt{|z - y|} $
I arrived at the relation
$\sqrt x + \sqrt y \geq \sqrt{x+y} $
But how the another variable $z$ comes into the picture remains question to me. Any help appreciated.
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
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 $|x-y|=|(x-z)+(z-y)|\leq |x-z|+|z-x|.$
