Showing the convergence of a sequence of compact nonempty sets

Question

Given convergent sequences of compact sets $\{A_k\}$ and $\{B_k\}$ with $\lim_{k \rightarrow \infty} A_k = A_{\infty}$ and $\lim_{k \rightarrow \infty} B_k = B_{\infty}$, $A_k \cap B_k \neq \emptyset \; \forall k,$ show that $A_{\infty} \cap B_{\infty} \neq \emptyset$

Answer 1

For $k\in\Bbb N$ let $x_k\in A_k\cap B_k$. For any reasonable definition of convergence of a sequence of compact sets is, you should be able to show that some subsequence $\langle x_{n_k}:k\in\Bbb N\rangle$ of $\langle x_n:n\in\Bbb N\rangle$ converges to a point $a\in A_\infty$. Then use the same argument to show that some subsequence of $\langle x_{n_k}:k\in\Bbb N\rangle$ converges to a point $b\in B_\infty$. Assuming that your spaces are Hausdorff, $a=b\in A_\infty\cap B_\infty$. The actual details, however, will depend on your definition of convergence of a sequence of compact sets, which you’ve not given us.