Approximating points of a Banach space by points of an operator image

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 $({\epsilon }_n{)}_{n=1}^{\mathrm{\infty }}$ of positive real numbers with ${\lim }_{n\to \mathrm{\infty }}{\epsilon }_n=0$, the set $A={x\in X:\forall n\in\mathbb N;\exists a\in A_n;;|x-a|< \varepsilon_n}$ is nowhere dense in $X$.