Let $X$ be a non-empty complete metric space. Give an example to show that the intersection of a descending sequence of non-empty closed subsets of $X, A_{1} \supset A_{2} \supset \cdots$, can be empty. Show that if we also assume that

$$\lim _{n \rightarrow \infty} \operatorname{diam}\left(A_{n}\right)=0$$

then the intersection is not empty. Here the diameter $\operatorname{diam}(A)$ is defined as the supremum of the distances between any two points of a set $A$.

We say that a subset $A$ of $X$ is dense if it has nonempty intersection with every nonempty open subset of $X$. Let $U_{1}, U_{2}, \ldots$ be any sequence of dense open subsets of $X$. Show that the intersection $\bigcap_{n=1}^{\infty} U_{n}$ is not empty.

[Hint: Look for a descending sequence of subsets $A_{1} \supset A_{2} \supset \cdots$, with $A_{i} \subset U_{i}$, such that the previous part of this problem applies.]