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  • My idea is essentially showing that a subset such as in theorem is totally bounded. (since in this way the result follows by well-established equivalence between totally boundeness and relative compactness in metric spaces.) Thus, chosen $\epsilon >0$, I was looking for a finite collection $S$ of vectors in $\ell^p$ such that every vector in $K$ is far from at last one vector in $S$ less than $\ep
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