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The collection $F$ of open intervals of the form $(1/n,2/n)$ where $n = 2,3,\ldots$ is an open covering of the open interval $(0,1)$.

Prove that no finite subcollection of $F$ covers $(0,1)$ without using the fact it isn't compact.

Suppose there exists a finite subcollection of $F$ that covers $(0,1)$. By Heine-Borel, there exist a countable collection of $F$ such that

$$

(0,1) = \bigcup_{n = 1}^mI_n

$$

How should I continue on?

Prove that no finite subcollection of $F$ covers $(0,1)$ without using the fact it isn't compact.

Suppose there exists a finite subcollection of $F$ that covers $(0,1)$. By Heine-Borel, there exist a countable collection of $F$ such that

$$

(0,1) = \bigcup_{n = 1}^mI_n

$$

How should I continue on?

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