Let $K$ and $L$ be nonempty compact sets, and define
$$d = \inf\{ |x-y| : x \in K \textrm{ and } y\in L \}$$
This turns out to be a reasonable definition for the distance between $K$ and $L$.
- If $K$ and $L$ are disjoint, show $d > 0$ and that $d = | x_{0}-y_{0}|$ for some $x_{0}\in K$ and $y_{0}\in L$;
- Show that it's possible to have $d = 0$ if we assume only that the disjoint sets $K$ and $L$ are closed.
My teacher explained to me that I could say something along the lines of: If we find a sequence in the set $K$, $(x_n)$, and a sequence in the set $L$, $(y_n)$ and we assume that they have the same limit then they are approaching the same number. But since $K$ and $L$ are compact, then they are closed which means they contain their limit points and $K$ and $L$ must contain the limit of the sequences. I am not really sure where to go from there or how to write that in a formal proof. Any help would be appreciated!
$$d=\lim_{n \to \infty}|x_n-y_n|.$$
Since $K$ is compact, there is a convergent subsequence $(x_{n_k})$ with $x_{n_k} \to x_0 \in K$ ($k \to \infty$)
Since $L$ is compact, $(y_{n_k})$ contains a convergent subsequence $(y_{n_{k_j}})$ such that $y_{n_{k_j}} \to y_0 \in L$ ($j \to \infty$).
Then we have: $$ d=\lim_{j \to \infty}|x_{n_{k_j}}-y_{n_{k_j}}|=|x_0-y_0|.$$