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Convergent sequences are always Cauchy; for metric spaces, compactness and sequential compactness are the same.
by dbfclark 2y ago
Convergent sequences are always Cauchy; for metric spaces, compactness and sequential compactness are the same.
- Smaug123 2y agoI think the question is more like: > Turing says that a certain space, the space of all compact subsets of [0,1]^2 endowed with the metric "integral of minimal distance required to transform {1 ink at each point of P1} u {infinite amount of ink at (2, 0)} into {1 ink at each point of P2} u {infinite amount of ink at (2,0)}", is conditionally-compact. How is that related to the article's argument? This is not obvious, I think. The article has moved away from Turing's "integral of the distance we have to transfer ink", instead using "maximum distance we have to transfer any ink", and I don't have a great intuition for whether this is a legit transformation of the argument. (I'm sure both proofs are correct, but it's not obvious to me that they are the same proof.)
- tempfile 2y agoYes and yes, but conditional compactness is different, and Cauchy sequences are not always convergent. That's why I mentioned completeness.