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Show that if K is a compact set in $\mathbb{R}^n$, and U is an open set such that K $\subset$ U, then there exist $r>0$ and a finite collection of disjoint balls $\{B(x_j,r)\}_{j=1}^{N}$ such that $K \subset \cup_{j=1}^{N}B(x_j,3r) \subset U$.

All I know is since K is compact hence there exist finite points $\{x_1, x_2,x_3,...x_n\}$ such that $K= \cup_{i=1}^{n}B(x_i,r_i).$

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Hint:

a) Choose $r < \min {r_i}$. This $\min$ can be chosen since $N$ is finite.

b) then $\{B(x_j,r)\}_{j=1}^{N}$ is a set of disjoint open sets.

c) $r$ should be chosen such that $3r > r_k$ for some $k$

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    $\begingroup$ can you elaborate point (c) please. $\endgroup$ – silent learner Jun 10 '15 at 8:15

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