Let $x, y, z$ be some three (real-valued) time series. Prove that $$\sqrt{1/2-\rho(x,z)/2} \leq \sqrt{1/2-\rho(x,y)/2} + \sqrt{1/2-\rho(y,z)/2}$$
After this assuming that two time series cannot be perfectly correlated if they are not equal it will be possible to show that $d = \sqrt{1/2-\rho/2}$ is indeed a metric.
To be honest, this exercise was given in de Prado's book called "Machine Learning for Asset Managers" as an exercise for a reader, i.e. we know for a fact that $d$ is a metric, however, I am struggling with proving aforementioned triangle inequality. If not the full solution, can one share an idea to solve this without brude forcing correlation formulas into the inequality? I tried that and could not finish till the end.
If that's inevitable, it would help so much if one provided the proof.
With kind regards, Nura