Give an example of an open cover of the segment $(0,1)$ which has no finite subcover.
Example: Taking $G_n=(0,1-1/n)$ for $n>1$. It is obvious that $(0,1)\subset \cup_{n=2}^{\infty}G_n$ but $\{G_n\}$ has no finite subcover of $(0,1)$.
Does this example satisfy the conditions of problem?