Question

Prove by using the triangle inequality that |2 arctan(2x) + αx5 | ≤ (α + 4)|x| if |x| ≤ 1

Answer 1

I have to show that d is a metric on the real numbers, and the first three axioms are straight forward, the triangle inequality poses a problem. I know we need to get

$d(x,y)=arctan|x−y|≤arctan|x−z|+arctan|z−y|=d(x,z)+d(z,y),$

so what I've tried is

$arctan|x−y|=arctan|x−z+z−y|≤arctan(|x−z|+|z−y|),$

but I'm not even sure if this accomplishes anything because I don't know how to split it up.