I'm trying to prove that interval $(0,1]$ is not compact by showing it doesn't have Heine-borel property.
I know a set is compact if a set is closed and bounded or has BW property or has Heine-borel property. But I'm trying to use heine-borel property to prove that it is not compact. I know I have to use the definition of open cover to prove this, but I don't know how to begin.
my guess: in order to prove $(0,1]$ is not compact by showing it doesn't have heine-borel property, is to show that there exists open cover $(0,1]$ that cannot be reduced to a finite subcover. but then what would be $\mathscr{U}$?
Best Answer
You want to construct an open cover for which no finite subcover can still cover all of the interval $(0,1]$. One way you might do this is to take a collection of covers $U_n = (a_n, 2)$, where $a_n \to 0$.
If you create $a_n$ such that all $a_n > 0$, then clearly no finite subcover will still cover the interval.