# Show (0,1) is not compact [duplicate]

Let $I_n=\left(\frac{1}{n},1\right)$. Show that $(0,1)$ is not compact: show that any finite collection of $\{I_n\}$ will not cover $(0,1)$.

Give me a hint.

• This is a straightforward proof. Take any finite subset of $\{I_n\}$ and exhibit an element of $(0,1)$ not contained in any of the selected $I_n$. Mar 11 '15 at 22:34

Any finite collection out of your set will have an element with the greatest $n$
• You just have to find one point that is in $(0,1)$ but not contained in one of the intervals in the finite collection. That proves this finite collection is not a cover of $(0,1)$. Then you argue you can do that no matter what finite collection you are given. Mar 11 '15 at 22:39