Strongly equivalent metrics are equivalent

I have $d,d'$ metrics in X and that they are strongly equivalent. In my case, this means that:$\exists\alpha,\beta\in\mathbb{R}_{++}$ so that $\alpha d<d'<\beta d$I want to show that they are equivalent by proving that:$\forall x\in X,\forall\epsilon>0,\exists\delta>0:B_d(x,\delta)\in B_{d'}(x,\epsilon)$and vice-versa.My argument is: take $x\in X,\epsilon>0$ and build the ball $B_{d'}(x,\epsilon)$. If we take the ball $B_{d}(x,\frac{\epsilon}{\alpha})$, then :$$y\in B_{d}(x,\frac{\epsilon}{\alpha})\Rightarrow d(x,y)<\epsilon/\...Read more