Consider a sequence $(A_n)$ of bounded, convex subsets of $\mathbb{R}^n$ such that $\text{int}(A_n)\neq\emptyset$ for all $n$ and $\text{clo}(A_n)\neq \text{clo}(A_m)$ whenever $m\neq n$, where $\text{int}(\cdot)$ and $\text{clo}(\cdot)$ denotes interior and closure, respectively.
Suppose $A_n\rightarrow A$ in the Hausdorff metric.
Is it true that every sequence $(x_n)$ in $\mathbb{R}^n$ such that $x_n\in A_n$ for all $n$ converges to a point in $A$?
No not quite. The sequence may jump around within $A_n$. E.g. A_n a line interval that rotates to a fixed interval $A$. And $x_n$ jumps up and down the interval as n increases. You'd have to pass to a subsequence of $x_n$ to get convergence.