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For a compact set $K\subset \Bbb R^n $ and $\delta>0$ show that that there exists a finite number of elements in $K$, say $x_1,x_2,\dots,x_k$ such that any other element $x$ of $K$ is at a distance of less than $\delta$ from at least one of the elements $x_1,x_2,\dots,x_k$.

To prove this, I'm trying to show that $\|x-x_j\|<\delta$ for at least some $x_{j}$ $(j=1,\dots,k)$. But I haven't succeeded so far.

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    $\begingroup$ What is your definition of being compact? $\endgroup$ – Damien L Sep 17 '14 at 20:44
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For $x\in K$, put $B_\delta(x) = \{y\in K:\, |y - x| < \delta\}$. Since $x\in B_\delta(x)$, the open sets $B_\delta(x)\subset K$ for $x\in K$ cover $K$ and thus admit a finite subcover $\{B_\delta(x_1), \dots, B_\delta(x_k)\}$.

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Let $\delta>0$ be given. Clearly the family of open balls $\{B(x, \delta)\}_{x \in K}$ covers $K$. Since $K$ is compact, finitely many of the balls say $\{B(x_n, \delta)\}_{n=1}^k$ covers $K$.

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