assumed distance in metric spaces 
Determine whether or not the distance between nonempty $A,B\subset X$ for metric space $X$ is assumed if A and B are closed.

The definition of distance between sets A and B is $d(A,B)=\inf\{d(a,b)|a\in A, b\in B\}$ and the distance is "assumed" if $\exists a_o\in A, b_o\in B$ : $d(A,B)=d(a_o,b_o)$. 
I am trying to see how it could be the case that $A,B$ closed $\implies$ $d(A,B)$ assumed. Is it because closed sets contain all their accumulation points?

It is not the case.  Consider the two curves in $\mathbf{R}^2$, $A=\{(x,-1/x)\mid x<0\}$ and $B=\{(x,1/x)\mid x>0\}$.  $d(A,B)=0$ but $d(u,v)>0$ for all $u\in A$ and $v \in B$.
