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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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