Let's consider a set $X$ with two different metrics (distance function) $d_1, d_2$ on $X$.
Is $\lim_{n\to\infty} d_1(x_n,x)=0 $ equivalent to $\lim_{n\to\infty} d_2(x_n,x)=0$?
I mean, when we can define two different metrics on a set, do the two different metrics give the same limit relation on the set?