How to show that the natural logarithm is Lipschitz on $[\beta, \infty)$ I want to show the following result:

Let $\ln(x)$ have domain $D = [\beta, \infty)$ then $|\ln(x) - \ln(y)|
 \leq \dfrac{1}{\beta} |x-y|, \forall x,y \in D$

I am confused as to how to prove this seemingly simple looking claim.
Can someone please help?

The derivative, being $1/x$ is upper bounded by $1/\beta$ and also monotone decreasing on this domain.
