  Answering one problem that has its origins in quantum mechanics, we prove
that for any sequence $(A_n)_{n\in\mathbb N}$ of convex nowhere dense sets in a
Banach space $X$ and any sequence $(\varepsilon_n)_{n=1}^\infty$ of positive
real numbers with $\lim_{n\to\infty}\varepsilon_n=0$, the set $A=\{x\in
X:\forall n\in\mathbb N\;\exists a\in A_n\;\;\|x-a\|< \varepsilon_n\}$ is
nowhere dense in $X$.

Functional Analysis