(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Suppose [itex](X,d)[/itex] is a metric space, and suppose that [itex]A,B\subseteq X[/itex]. Show that [itex]dist(A,B)=dist(cl(A),cl(B))[/itex].

2. Relevant equations

[itex]cl(A)=\partial A\cup A[/itex].

[itex]dist(A,B)=\inf \{d(a,b):a\in A,b\in B\}[/itex]

3. The attempt at a solution

Its clear that [itex]dist(cl(A),cl(B))\leq \min\{dist(A,B),dist(A,\partial B),dist(\partial A,B),dist(\partial A,\partial B)\}\leq dist(A,B)[/itex]. I just can't exactly find a way to go the other direction of the inequality. I'm not looking for a direct solution, just some intuition.

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# Homework Help: Distance between Sets and their Closures

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