I have 2 distance functions $d(x,y)=|x^2-y^2|$ and $d(x,y)=|x^3-y^3|$ and I am trying to prove that they are metrics on $\mathbb R$, or give a counterexample that they are not metrics on $\mathbb R$.
I managed to prove the first 3 properties for the 2nd function namely:
$d(x,y)\ge0$
$d(x,y)=0$ iff $x=y$
$d(x,y)=d(y,x)$
and I now have shown that the 1st function is not a metric.
But for the triangle inequality property, $d(x,y)\le d(x,z) + d(z,y)$, Im stuck on it. How would go about proving it for $d(x,y)=|x^3-y^3|$? Thanks.