Distance between sets implies completness of a metric space 
Given the metric space $(X,d)$ suppose that for every nonempty, closed $A,B\subseteq X$ with $$A\cap B=\varnothing, \mathrm{dist}(A, B)>0.$$
  Show that  $(X,d)$ is complete.

But given the definition of distance as $$\operatorname{dist}(A,B)=\inf\{d(a,b):a \in A,b \in B\}$$ 
and given that  $$A\cap B=\varnothing$$ 
and the fact that $A,B$ are closed 
we have that  $$\mathrm{dist}(A, B)=0\iff A\cap B \neq \varnothing$$


Am I missing something?

I don't think this statement is true.
For example, the interval $(0,1] \in \mathbb{R}$ seems to has the property from the hypothesis but clearly is not complete.
But I'm not sure if the property is really true on this space.
