Let $(M,d)$ be a metric space and let $H(M)$ denote the set of closed and bounded subset in $M$. Then $(H(M),d_H)$ is a metric space where $d_H$ denotes the Hausdorff distance. Let $\chi$ be the Hausdorff (or ball) measure of non-compactness
How do I prove that the map $\chi\colon H(M)\to\mathbf{R}:B\mapsto\chi(B)$ is Lipschitz?