# Lipschitz continuity for an iterated function system

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 $K\geq L$?

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The metric $\rho$ is known to be a Hausdorff distance –  Ilya Apr 3 '12 at 19:56
Note that $M$ and $N$ embed isometrically into $CB(M)$ and $CB(N)$ and that $F$ is an extension of $f$. –  t.b. Apr 4 '12 at 4:21