I need to find an open cover of {x: x>0} with no finite subcover.
Frankly, a simple answer is best.
It did ask for an open covering: $\displaystyle \left\{ {\left( {\frac{1}{n},\infty } \right):\,n \in \mathbb{Z}^ + } \right\}$.
This makes it very clear the you understand the thrust of the question.