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show that $$\{ 1/n : n\in \Bbb N\}$$ is not compact using open covers


given any open cover then exists a finite subcover would mean that it is compact.

So it should be enough to say that for some open cover there is no finite subcover

intiution is to place the collection of open balls at points $1/n$ with some radius $\epsilon$ want to let epsilon the distance bw $1/{n}$ and $1/{n-1}$ and somehow show that is not finite since n goes for ever but feel kind of lost

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Choose open balls around each $\frac{1}{n}$ with radius $\varepsilon<\frac{1}{2}min\{\frac{1}{n}-\frac{1}{n+1},\frac{1}{n-1}-\frac{1}{n}\}$. Then obviously these balls are disjoint and each ball contains only one number of the form $\frac{1}{n}$. But the latter set is infinite, so having a finite subcover would yield a contradiction.

Thus, this set is not compact.

QED.

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