# Answer to Question #17326 in Real Analysis for Justin

Question #17326

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.

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.

Expert's answer

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.

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.

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