Metric Spaces (Extreme Value Theorem)
Solution 1:
I think the argument of choosing a sequence $\{ x_n \}$, that is your paragraph,
... for some $r>0$, it was possible to inductively create a sequence $\{ x_n \}_{n=1} ^\infty$ for which $$ \forall n \in \mathbb{N}\, , x_n \not \in B(x_i, r)\, , \quad \forall i = 1, \dots, n-1 $$
still works, except that after $\{ x_n \}$ has been chosen, we instead consider the collection of open balls $\{ B(x_n, \frac{r}{3})\}$. This collection will be disjoint, since any two centers are at least $r$ unit of distance apart.