Let $\{E_n\}$ be a collection of bounded and closed subsets in a metric space $X$ such that $E_{n+1} \subset E_n$ and $lim_{n\to\infty} diam E_n = 0$.
It's a theorem that if $X$ is complete, then $\bigcap_{n\in \mathbb{N}} E_n$ is a singleton.
However, i have proved that $\bigcap_{n\in \mathbb{N}} E_n$ is a singleton where $Int(E_n) ≠\emptyset$ even if $X$ is not complete, but just an arbitrary metric space.
I don't actually believe my proof.. Is it true? Or please give me a counterexample!