# exception for disproving this sufficient condition

The following is sufficient but not necessary condition for topological equivalence:

for each x $\in$ X, there exist positive constants $\alpha$ and $\beta$ such that, for every point y $\in$ X $\alpha d_{1} (x, y) \leq d_{2} (x, y) \leq \beta d_{1} (x, y)$

I am trying to find some exception where above one is true but metrics are not equivalent especially in $R$ domain

$$d(x,y)=\frac{|x-y|}{1+|x-y|}$$
is a metric on $\Bbb R$ that generates the usual topology, but there is no $\beta>0$ such that $$|x-y|\le d(x,y)$$ for all $x,y\in\Bbb R$: for any positive integer $n$ we have
$$\frac{|n-0|}{d(n,0)}=\frac{n}{\frac{n}{1+n}}=1+n\;.$$