Suppose A and B are nonempty closed bounded subsets of $\displaystyle \mathbb{C}$.

1. Show that there exist $\displaystyle a_0 \in A, b_0 \in B$ such that $\displaystyle |a_0-b_0|=\inf\{|a-b|:a\in A, b\in B\}$.

2. Show that this is not true if the boundedness condition on $\displaystyle A$ and $\displaystyle B$ is dropped.

3. What if only one of $\displaystyle A$ or $\displaystyle B$ is bounded but both are still closed?

Question about #1, if $\displaystyle A, B$ are closed and bounded, does this mean every sequence in $\displaystyle A, B$ is bounded?

Also, I'm clueless about #2 and #3. Some help please.