An example of an infinite open cover of the interval (0,1) that has no finite subcover I've been having a hard time solving this problem that I was given in class. The problem states
" Give an example of an infinite open cover of the interval (0,1) that has no finite subcover." 
I know that the set has to be compact and that both 0 and 1 are limit points. Aside from those two known factors I'm at a complete loss as to how to go about solving this problem.

In case you are still interested. Abbot's has this problem as Example 3.3.7. For each point $x\in(0,1)$ consider the open interval $0_x = (x/2,1)$. The infinite union of $O_x$'s forms an open cover of $(0,1)$. Take any finite subset of such $O_x$, and set $x'=\operatorname{min}(x_1,x_2,\dots,x_n)$. You can still find a number $0<y\leq x'/2$ not in any subfinite union.
