The following question popped up while thinking about Cantor's intersection theorem.
Consider a metric space $X$ so that we may talk of the diameters $\delta(A)$'s of its subsets $A$'s. Let $A_1\supseteq A_2\supseteq\cdots$ be a nested sequence of subsets of $X$ with $A_1$ bounded (that is, $\delta(A_1) < +\infty$). Then, can we say that $\delta(\cap_i A_i) = \lim_i\delta(A_i)$ (as intuition so loudly screams)?
The answer is affirmative if $A_i$'s are all compact (as shown by Brian M. Scott here). Cantor's intersection theorem says that this is true if $A_i$'s are closed with diameters going to zero. But can nothing more be said in general? Counterexamples are welcome.
Let $A_1=\{e_i: i\ge 1\}$ be a countable orthonormal basis of an infinite-dimensional Hilbert space. Take $A_n:=\{e_i: i\ge n\}$. Then all these sets have the same diameter $\sqrt 2$ while their intersection is empty and hence has zero diameter.