If ${A_n}$ is a sequence of nonempty closed convex sets in a Banach space such that $A_{n+1} \subset A_n$ for all $n \in \mathbb{N}$, must $\bigcap_{n \in \mathbb{N}} A_n$ be nonempty?
My Answer: Counterexample: Consider the sets $A_n$ in $\mathbb{R}$ defined by:
$A_n =[0,1/n]$
Each set $A_n$ is nonempty, closed, and convex. However, the intersection of all $A_n$ is empty:
$⋂_{n=1}^∞ A_n =∅$
Thus, the intersection property does not always hold, and there exists a counterexample where the intersection is empty.
Is this correct? Any help is appreciated!
Your counterexample doesn't work as explained in the comments.
For a counterexample that does work, take $B_n = [n, \infty) \subseteq \Bbb{R}$. Then, for all $n$, $B_n$ is closed and convex and $B_{n+1} \subset B_n$, but $\bigcap_nB_n = \emptyset$.
The $B_n$ above are closed, but not compact. A nest of compact subsets of any Banach space must have a non-empty intersection.