The assumption of nonemptiness in the theorem (3.10 from Rudin) about the intersection of nested compact sets

Question

There is Theorem 3.10(b) in baby Rudin.

If $K_n$ is a sequence of compact sets in a metric space $X$ such that $K_n\supset K_{n+1}$ and if $$\lim_{n\to \infty}\text{diam}K_n=0,$$ then $\bigcap_{n=1}^{\infty}K_n$ consists of exactly one point.

I think that all $K_n$ must be a nonempty. But why Rudin didn't write this?

Answer 1

As many authors, Rudin defines diameter only for nonempty set; thus, making an assumption about $\operatorname{diam}K_n$ means that $K_n$ is implicitly assumed nonempty.

A related question: What is the best way to define the diameter of the empty subset of a metric space?