I'm having problems wrapping my head around the part with $\rho_i$.Here goes:
$A \subset \mathbb{R}^n$ is compact, $\rho$ is a positive real-valued function defined on $A$. Prove: $\exists$ finitely many points $\{x_1, x_2, ..., x_n\} \subset A$ such that
a) $B_{\rho_i}(x_i) \bigcap B_{\rho_j}(x_j) = \emptyset, i \ne j; \rho_i = \rho(x_i) $
b) $A \subset \bigcup^{N}_{i=1} B_{2\rho_i}(x_i).$ (For some $N$)
Since $\rho$ seems to be arbitrary, I don't see what should stop $min_{x_i \in A} \{\rho(x_i)\}$ from potentially getting too big for a) to hold. Also I see that obviously there is an open subcovering for A, but why exactly as stated in b)?