What is an example of nested bounded convex and closet sequence of set in a Banach space with empty intersection?

Question

What is an example of nested bounded convex and closet sequence of set in a Banach space with empty intersection?

I cannot imagine an example of that. Thanks.

Answer 1

To give a concrete example (based on Daniel Fischer's answer), consider $c_0$ (the space of sequences $x_1,x_2,\dots$ converging to $0$, with the supremum norm), and let $$ A_n = \{x\in c_0 : \|x\|\le 1,\ \sum_{k=1}^\infty 2^{-k}x_k\ge 1-1/n \} $$ This is evidently a nested sequence of nonempty closed convex sets. Yet, $\bigcap A_n$ is empty, since the only element it could conceivably contain, $(1,1,1,\dots)$ is not in $c_0$.