I tried to solve this problem:
Let $(X,d)$ a metric space. Show that $d$ and $\bar{d} =\min({d(x,y),1})$ are topologically equivalent metrics.
I proved that $\bar{d}$ is a distance, then I tried to show that every open ball on $(X,d)$ is contained on an open ball on $(X, \bar{d})$ and vice versa.
if $r<1$, and $\forall x \in X$ $B_r (x)=\bar{B}_r (x)$.
But if $r \ge1$, $\bar{B}_r (x)=X$, then I can't find a ball on $(X,d)$ such that $\bar{B}_r (x) \subseteq B_{r'} (x)$