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I am wondering if we can assume the following if your function $f$ is $K-$lipschitz: $$\delta(f(A),f(B)) \leq K*\delta(A,B),$$ where we use the definition: $$\delta(A,B) = \inf\{\epsilon \geq 0\,|\ A \subseteq B_\epsilon \text{ and } B \subseteq A_\epsilon\}.$$ I have troubles to see how one should get this.. Any help would be great!

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The more slick notation may be tripping you up here. Denote: $$h(A, B) = \sup_{a \in A} \inf_{b \in B} d(a, b) = \inf\{\varepsilon \ge 0\ : A \subseteq B_\varepsilon\}.$$ Then $\delta(A, B) = \min\{h(A, B), h(B, A)\}$. We have, \begin{align*} h(f(A), f(B)) &= \sup_{a' \in f(A)} \inf_{b' \in f(B)} d(a', b') \\ &= \sup_{a \in A} \inf_{b \in B} d(f(a), f(b)) \\ &\le \sup_{a \in A} \inf_{b \in B} Kd(a, b) \\ &= K\sup_{a \in A} \inf_{b \in B} d(a, b) \\ &= Kh(A, B). \end{align*} So, $$\delta(f(A), f(B)) = \min\{h(f(A), f(B)), h(f(B), f(A))\} \le \min\{Kh(A, B), Kh(B, A)\} = K \delta(A, B).$$

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