Prove that interval $(0,1]$ is not compact

Question

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 $\mathscr{U}$?

Answer 1

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.

Answer 2

Define $I_{n} = (\frac{1}{n}, 2)$

Then $$ \bigcup_{n \in \mathbb{N}} I_{n} $$ covers $(0,1]$ but it's easy to see that finitely many $I_{n}$ can't cover the interval. This is because if a finite cover $\{I_{n}\}_{n=1}^{k}$ existed, then this would not cover $(0, \frac{1}{k})$ and thus wouldn't be a cover at all.