Given $x \in \mathbb{R}^n$, $x \neq 0$, let $x' = x/|x|$ (where $|\cdot|$ is the euclidean length) be its projection onto the unit sphere. I would like to prove that $$ |x' - y'| \leq 2 \max(1/|x|,1/|y|) |x-y|. $$ Is there some nice way to do that? Analytic or maybe geometric?
This came up while I was reading some harmonic analysis text and I would like to use it to check the Hormander condition in the Calderon-Zygmund theorem.