Let $X$ be a Banach space, and $B_1\supseteq B_2 \supseteq\cdots $ . Show that $\bigcap\limits_{i=1}^\infty B_i\neq\emptyset$

Question

Let $X$ be a Banach space, and $B_1\supseteq B_2 \supseteq \cdots $ a sequence of closed balls with radius $r_i$ and center $x_i$. Show that $$\bigcap_{i=1}^\infty B_i\neq\emptyset$$

I proved that $r_i\leq r_j$ for $B_i\subseteq B_j$ and $r_i\rightarrow\alpha\geq0$ so, for $\varepsilon>0\ \exists N\in\mathbb{N}$ such that $|r_i-r_j|<\varepsilon$ for $i,j>N$ and just left to show that $\|x_i-x_j\|<\varepsilon$ so the centers is a chauchy sequence and it converges, so my question is how can I prove that $\|x_i-x_j\|<\varepsilon$?