Setup: Let $B=\text{conv}\{x_1,..,x_k\}$ be the convex hull of $k$ points in $\mathbb{R}^m$. Consider $k$ sequences $(x_i^n)_{n\in{\mathbb{N}}}$ converging to the corners $x_i^n\rightarrow x_i$ for $i=1,..,k$ and the corresponding convex hulls $B^n=\text{conv}\{x^n_1,..,x^n_k\}$. We also assume that $B$ has a non-empty interior $int(B)\neq\emptyset$ in $\mathbb{R}^m$ ($B$ has $n$-dimensional volume).
Question: I want to prove that $\forall x\in int(B)$ we will have $ x \in B^n$ eventually. This seems intuitive, but I'm looking for a proof.