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: $$exists sge 1:quad forall x,y,zin X:quad d(x,z)le sbig(d(x,y)+d(y,z)big).$$

MathOverflow Asked by Seddik Merdaci on January 4, 2021

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