Does the lemma remain valid in b-metric space?

Let $(X,d)$ be a complete metric space.

$$CB(X)={A : A text{ is a nonempty closed and bounded subset of }X },$$
$$D(A,B)=inf {d(a,b) : ain A , bin B},$$
$$sigma (A,B)=sup {d(a,b) : ain A , bin B},$$
$$H(A,B)=max {sup_{xin B} D(x,A) , sup_{xin A} D(x,B)}.$$

Lemma:
Let $A,Bin CB(X)$, and let $xin A$. Then, for each $alpha>0$, there exists a $yin B$ such that
begin{equation}
d(x,y)leq H(A,B)+alpha.
end{equation}

Question : How can we prove the lemma, and that this lemma remains valid in b-metric space?

A b-metric space means the same as metric space, with triangle inequality replaced with: $$exists sge 1:quad forall x,y,zin X:quad d(x,z)le sbig(d(x,y)+d(y,z)big).$$