In a complete metric space $(X, d)$, if a sequence of open balls $\{B(x_i, r_i)\}_{i=1}^\infty$ satisfies

$$ \exists \epsilon > 0 ~~s.t.~ B(x_{i+1}, (1+\epsilon)r_{i+1}) \subset B(x_i, r_i), \forall i \ge 1 \tag{1} $$

then $\bigcap_{i=1}^\infty B(x_i, r_i) \neq \emptyset$.

Here are several related facts:

1. If we further require that $r_i \rightarrow 0$ as $i\rightarrow \infty$, then $\bigcap_{i=1}^\infty B(x_i, r_i) \neq \emptyset$ holds.

2. There is a example shows that $\bigcap_{i=1}^\infty B(x_i, r_i) = \emptyset$ if we replace $(1)$ by $\overline{B(x_{i+1}, r_{i+1})} \subset B(x_i, r_i)$ for all $i\ge 1$ where $\overline{B}$ is the closure of $B$

3. In a complete metric space, $B(x, r) \subset B(x', r')$ may hold for $r > r'$.