I'm currently reading the book Introduction to Topology by Gamelin. There is a problem on the first chapter that I could not figure out. Could anyone give me some hints please?
Two metrics $d,p$ on $X$ are equivalent if they determine the same open subsets. Show that two metrics $d,p$ on $X$ are equivalent if and only if the convergent sequences in $(X,d)$ are the same as the convergent sequences in $(X,p)$.
Thank you very much.