Question

Suppose that K1,K2 c R^n are nonempty compact sets. Let
d(K1;K2) = inf{d(x1, x2)| (x1 cK1) and (x2c K2)}:
Show that d(K1,K2) = 0 implies( K1 n K2) is not empty set.

Answer 1

Since they are compact in R^nthen they are bounded and closed.
As they are compact then function d: K1 times K2 -> R defined over again
compact set obtains its minimal value.
Since d(K1,K2)=0 then inf is actually min, and minimal value is 0. Since for
some point (x1,x2) this minimum occures then
d(x1,x2)=0 at this point. Then by definition of metrics d we have x1=x2, and
LHS is in K1, and RHS is in K2. Then K1 and K2 have common point, and thus their intersection is nonempty.