Compactness hypothesis in Heine's theorem

Question

Here is the proof from Apostol's analysis

I think the main point in the proof is the existence of a smallest radius among $\frac{r_1}{2}\ ,\ \frac{r_2}{2}......,\frac{r_k}{2}$, but I don't see how the compactness hypothesis is necessary for this. Suppose $A$ wasn't compact, then the set of radii in question may be infinite. However, this is a set of real numbers which are bounded below by zero and hence have an infimum which can be used in the rest of the argument and it will still be valid.

Answer 1

Infimum of an infinite sequence of positive numbers can be $0$ as in the case of $(\frac 1 n)$. You cannot have $\delta =0$ in the proof.