Let $(M,d_M)$ and $(N,d_N)$ be metric and $ CB(M)=\{\mbox{all closed bounded subsets of }M\}. $ Let $f: M\to N$ be a Lipschitz map with Lipschitz constant $L$. Define a map $ F:(CB(M),\rho)\to (CB(N),\rho) $ where $F(A)=\overline{f(A)}$ (a closure of $f(A)$). The metric $\rho$ is given by $\rho(A,B)=\max\{\sup_{a\in A}\inf_{b\in B}d(a,b),\sup_{b\in B}\inf_{a\in A}d(b,a)\}.$
From this, it can be shown that $F$ is Lipschitz with Lipschitz constant $K$ such that $K=L$. However, I have some trouble proving it. I have already shown that $K\leq L$. Can some help me show that $K\geq L$?