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Suppose $\{O_\alpha\}$ is an open cover of $[0,\infty)$. If the open interval $(1000,\infty)$ is one of the sets in the collection $\{O_\alpha\}$, show that there is a finite subcover for $[0,\infty)$.

I can't even begin this problem (never mind answer it).

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    What do you currently know about open covers and the existence of finite subcovers?2012-02-15

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Since $(1000, \infty)$ is one of the sets in your cover, you just need to find (finitely many) other sets in $\{O_\alpha\}$ to cover the rest of the interval, $[0, 1000]$. But this left over part is compact. (You are probably studying compactness now, so you should be able to take it from here.)

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    "You are probably studying compactness now". Yes, though the prof (and text) didn't call it that.2012-02-15