I am currently trying to show that the set $S = \mathbb{Q} \cap [0, 1]$ is not compact by showing the cover$$\mathcal{C} = \left\{ \left( \frac{p}{q} - \frac{1}{10^q!}, \frac{p}{q} + \frac{1}{10^q!} \right) \right\}_{p/q \in S}$$admits no finite subcover, with the assumption that each $\frac{p}{q} \in S$ is written such that $\gcd(p, q) = 1$.
I know there are far easier methods (contradict closedness with an irrational limit point) and covers to try this with, but I am very much so determined to do something with this specific cover.
So far, I have determined that for $a \in S$ to be contained within an interval for some other $b \in S$, the denominator of $a$ must be greater than that of $b$, with the whole idea being that if a theoretical finite subcover does exist, it always requires at least one more to cover some endpoint of an interval, thus leading to a contradiction.
My main questions are:
- Is this even true for this cover?
- If so, are there any decent hints that could push me in the right direction?