I am trying to solve the following exercise:
If $U$ is an open subset of $\mathbb{R}^n$, show that there exists an increasing sequence $\{A_k\}^\infty_1$ of compact contented sets such that $U=\bigcup^\infty_{k=1}\ \mathrm{int}\ A_k$.
Hint: Each point of $U$ is contained in some closed ball which lies in $U$. Pick the sequence in such a way that $A_k$ is the union of $k$ closed balls.
It seems "increasing sequence" means that $A_k \subset A_{k+1}$ for all $k$.
Now, I think the hint says that $A_{k+1}$ should be $A_k \cup B_{k+1}$ for some suitable ball $B_{k+1}$, starting with $A_1 := B_1$. But I don't see a way to choose those balls so that eventually all points of $U$ are covered?