## Related questions with answers

If

$K_1\ and\ K_2$

are disjoint nonempty compact sets, show that there exist

$k _ { i } \in K _ { i }$

such that

$0 < \left| k _ { 1 } - k _ { 2 } \right| = \inf \left\{ \left| x _ { 1 } - x _ { 2 } \right| : x _ { i} \in K _ { i } \right\}.$

Solution

VerifiedLet $K_1$ and $K_2$ be disjoint nonempty compact sets.

$K_i$ is compact. Let $(x_n)$ be a sequence such that $x_n\in K$ for every $n\in \mathbb{N}$. Because of Bolzano-Weierstrass Theorem 3.4.8. we know that there exists a subsequence $(x_{n_k})$ and since $K$ is closed it follows that the limit of $(x_{n_k})$ is in $K$.

In other words, every sequence in $K$ has a subsequence that converges to a point that is in $K$. Now, we know that $K_i$ is bounded wich means that it has an infimum and supremum.

Let $x_1\in K_1$ and $x_2\in K_2$ and lets observe their difference:

$|x_1-x_2| \text{ converges to } |K_1-K_2|$

As $K_1$ and $K_2$ are bounded it follows that $|K_1-K_2|$ is also bounded a therefore has infimum. We have:

$\boldsymbol{ |k_1-k_2|=\inf \{|x_1-x_2|\ ; \ x_i \in K_i\}}$

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