To show that an intersection of sets has a unique point

Question

Let $X$ be a non-empty complete metric space. Let $B_n=\{x\in X| \rho(x,x_n)< \epsilon_n\}$, where $\epsilon_n\to 0$ as $n\to \infty$. Let $B_n\supset B_{n+1}$. Prove that there exists a unique point in

$$\cap_{n=1}^{\infty}B_n$$

Where can I start this problem?

Answer 1

Hint: start by showing that $x_n$ is a Cauchy sequence.