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Let $K$ be a compact metric space. Let $\{U_{i}\}$ be an open covering of $K$. Prove that there exists a number $j>0$ such that any ball of radius $j$ is contained in some $U_{i}$.

Here is my attempt:

Let $\{U_{i}\}$ be a covering and let $\{U_{j}\}$ be a finite subcover. Let $j$ be the minimal radius from the balls of the finite subcover. This is the number we are looking for

Is it ok?

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    There is a nice short proof [here](http://mathblather.blogspot.com/2011/07/lebesgue-number-lemma-and-corollary.html).2012-09-02

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