Let $X$ be a countable set and $d_n,d$ locally finite metrics on $X$. Denote by $R_x^n$ (resp. $R_x$) the radius of the smallest closed ball in the metric $d_n$ (resp. $d$) about $x$ which contains at least two points.

Question: Suppose that $d_n\rightarrow d$ uniformly. Is it true that also $R_x^n\rightarrow R_x$ uniformly?

P.s. In case we can add the hypothesis that the $|C_n(x,R_x^n)|\leq C$, for a universal constant $C$ not depending either on $n$ on $x$ ($C_n(x,R)$ stands for the closed ball in the metric $d_n$ of radius $R$ about $x$).