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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?

1 Answers1

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

Robert Israel
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  • Since $\epsilon_n \to 0, \exists N$, s.t. $\rho(x_n,x_m )<\epsilon$ for a given $\epsilon$ and thus $\cap_n B_n = $limit point. –  Oct 27 '19 at 18:14