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I want to prove the Lebesgue number lemma:

Let $(X, d)$ be a compact metric space. Then given an open cover $\mathcal{A}$ of $X$, there exists $\delta \gt 0$ such that for each subset of $X$ having diameter less than $\delta$, there is an element of $\mathcal{A}$ containing it.

How can I prove this?

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    This is called the Lebesgue covering number, or the Lebesgue number of the cover: here is one proof: http://mathblather.blogspot.com/2011/07/lebesgue-number-lemma-and-corollary.htm2011-11-15
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    @gary: Thanks for the info...I think the link is broken though...2011-11-15
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    :the link was missing a single letter at the end; try this:http://mathblather.blogspot.com/2011/07/lebesgue-number-lemma-and-corollary.html2011-11-15
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    it's working now. Thanks.2011-11-15

4 Answers 4