Closed subsets of $\Bbb R^n$ that don't verify this property.

Question

I saw this property in an exercise about metric spaces: Let $(E,d)$ be a metric space and $A$, $B$ be two non empty subsets of $E$. If $A$ and $B$ are both compact then there exists $ (a,b) \in A \times B$ such that $d(a,b)=d(A,B)$.

The exercise then goes to show that if $E= \Bbb R^n$, A is compact and B is just closed, then the property also holds.

My question is, does there exists two unbounded closed subsets, $A$ and $B$, of $\Bbb R^n$ such that for every $ (a,b) \in A \times B$, we have $d(a,b)>d(A,B)$?

And if they do exist (which is probably the case because the exercice would mention it otherwise), an example in $\Bbb R$ would be much appreciated.

Answer 1

Let $$A=\bigcup_{i=1}^\infty [i-1+1/(10i),i-1/(10i)]$$ $$B=\mathbb Z$$ Then $d(A, B) =0$, but the two sets do not intersect.

Answer 2

Let $A$ be the set of all integers greater than $1,$ and let $B$ be $\lbrace n + \frac1 n \vert n \in A \rbrace.$

Answer 3

What about the graphs of $y=0$ and $y=e^{-x}$ on $[0,\infty)$?