Question: Assume $(x_n)$ is a weakly convergent sequence in an infinite dimensional Banach space $X$. Show that $\overline{\operatorname{conv}}\{x_n\}$ does not have any interior point.
The hint given tells me to assume $x_n \to 0$ $(weak)$, and that $K = \overline{\operatorname{conv}}\{x_n\}$ contains $0$ in its interior. But I don't understand how to go from the general case to this specific case.
I'm mainly interested in this hint rather than the question itself.
Thank you.