Exercise 15, § 3

Let $E$ be an infinite dimensional Banach space, $(a_n)_{n \geq 1}$ a countable total free family of points of $E$, and let $F_n$ be the $n$-dimensional subspace of $E$ generated by the $a_m$ for $m \leq n$. Let $S_n$ be the sphere with equation $\|x\| = n$ in $E$; in $S_n \cap F_n$ let $A_n$ be a finite set such that every point of $S_n \cap F_n$ is at a distance $\leq 1/n$ from $A_n$. Prove that $A = \bigcap_{n=1}^\infty A_n$ is such that its intersection with every closed bounded set is closed, but that $0$ is a limit point of $A$ for the weakened topology.