Let $H$ be compact in $\Bbb R^n$

Also assume that for every $x\in H$ there is an $r=r(x)$ such that $B\_r(x)\cap H=\{x\} $

Prove that $H$ is a finite set. 

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**Solution:**

Since $H$ is compact, $\exists$ a finite $$x\_1, x\_2, \dots , x\_N\in H \subseteq \bigcup B\_{r(x\_i)}$$

for $N\in\Bbb N$ 

Let's set $r:=$ min $\{r(x\_1),\dots , r(x\_N)\}$

Then, $$H \subseteq \bigcup \_{i=1}^{N} B\_r(x\_i)$$

Then, $$( \bigcup \_{i=1}^{N} B\_r(x\_i))\cap H= \{x\_1, \dots ,x\_N\}$$

So, $H=\{x\_1,\dots , x\_N\}$ is finite. 

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Does the proof of details true? Does there exist any drawbacks, missing in detail?

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