So I want to prove that for any $x,y \in \mathbb{R}$ the following inequality is true: $$|\ln(\frac{x+\sqrt{1+x^2}}{y+\sqrt{1+y^2}})| \leq |x-y|$$
So I know it's true if $x=y$, since it's $0\leq0$.
But I don't know how to evaluate for $x \neq y$.
I found this while going through calculus, so it should probably have a solution that uses calculus basics, I could be wrong though.
Any help would be appreciated.