Let $ f: \Re \rightarrow \Re$ be a continuous and strictly increasing function such that $f(0)=0$. Suppose $d_1$ and $d_2$ are two different metrics on the nonempty set $X$ and that $d_1(x,y)=f(d_2(x,y))$. Show that $(X,d_1)$ and $(X,d_2)$ have the same open sets.
$\textbf{My Attempt:}$
Let $r_1$ and $r_2$ be the collection of open subsets in the $d_1$ and $d_2$ metric. We want to show that $r_1=r_2$. So let $U \subseteq r_1$. I have seen, from an answer to a similar question on this website, that I must show for that for any $x \in U$ there exist $\epsilon >0$ such that the open ball $B_{d_2} (x,\epsilon) \subseteq U$. Also since $x \in U$ there exist $\delta > 0$ such that $B_{d_1} (x,\epsilon) \subseteq U$. I am quite lost as what to do from here on using that $f$ is a continuous bijection. We have not yet covered homeomorphisms in class so I will not be able to use any such theorems.
Also, sorry for not directly linking to the similar question I mention. I still need to look up how to do this.