Originally Posted by

**miatchguy** I'm trying to figure out a problem here and I don't quite get it.

Let $\displaystyle n \leftrightarrow G_{n}$ be a bijection between $\displaystyle N$ and $\displaystyle Q\cap[0, 1]$. For $\displaystyle \epsilon\in(0, 1)$, define $\displaystyle G_{n}$ to be the open set $\displaystyle B_{\frac{\epsilon}{2^n}}$-ball around the point $\displaystyle q_{n}$. Define $\displaystyle F=G^{c}\cap[0, 1]$. Show that F is compact.

Now I don't want the answer to this question; I want to figure it out. It's just that I could swear that the set $\displaystyle F$ is empty, which I know is compact. However, the next part of the question asks me to show that $\displaystyle 0<inf(F)$ and that $\displaystyle sup(F)<1$.

Basically, it looks to me like $\displaystyle G_{n}$ completely engulfs $\displaystyle [0, 1]$ and then some, so $\displaystyle G^{c}\cap[0, 1]$ wouldn't contain any values. Thus the set F is empty.