Let $\mathbf X = \Bbb R$ with distance function defined by $d(x,y) = {|x-y|}^\alpha$ , where $\alpha \in \Bbb R$ $(0<\alpha\le1)$.
Prove that $(\Bbb R , d)$ is a metric space.
The first three properties are easy, so I only need the triangle inequality. I tried to use Bernoulli's inequality, but did not lead me to the proof.
$d(x,y) \le d(x,z) + d(z,y)$
Please help me, Spanish is my language.
