How do I prove triangle inequality for this metric?

Question

I am not exactly sure how to prove triangle inequality for this metric. For other metrics the proof was more intuitive, I hope you can help me out.

$$d(x,y) = |x-y|^{3}$$

Answer 1

Consider $x=0, z=1, $ and $y=2$.

$d(x,y)=8, d(x,z)=1, $ and $d(y,z)=1$,

so the triangle inequality is not satisfied, and your $d(x,y)$ is not a metric.