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

Okay, so we've moved on from talking about R^n to talking about general metric spaces and the differences between the two. We're given that X (a metric space) satisfies the Bolzano-Weierstrass Property and that A and B are disjoint, compact subsets of X. Dist(A,B) is defined as the inf{d(x,y): x in A, y in B}. We're asked to show that Dist(A,B)>0.

2. Relevant equations

General theory of metric spaces: definition of a metric space, metric, etc.

3. The attempt at a solution

Okay, I think I must be missing something because to me it seems kind of trivial. My proof basically says that, by definition of a metric, d(x,y) is always greater than or equal to 0, with equality holding only when x=y. Since A and B are disjoint in our problem, x does not equal y for all x in A and y in B hence d(x,y) is always greater than 0. Since the inf is the min of these distances, it follows that the inf is always greater than 0.

I didn't use the fact that X satisfies the Bolzano-Weierstrass Property however, which makes me think that I'm missing something. Any help would be greatly appreciated.

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# Metric spaces and the distance between sets

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