Update: I've edited the question, since maybe it was a bit confusing and it's better to start with a more basic question.
I'm looking for properties of Gromov-Hausdorff convergence in the particular case when all metrics $d_n$ and $d$ are defined on a fixed finite or countable set $X$ and they give rise to locally finite spaces.
Fix a point $x\in X$ and 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. I suppose that every $(X,d_n)$ is $\delta_n$-connected; i.e. $\delta_n$ is a non-negative real number such that
$|R_x^n-R_y^n|\leq\delta_n$, for all $x,y\in X$
If $\delta'\geq0$ verifies $|R_x^n-R_y^n|\leq\delta'$, for all $x,y\in X$, then $\delta'\geq\delta_n$
For all $x,y\in X$, there is a $\delta_n$-connection; i.e. a finite sequence of points $x_0,x_1,\ldots,x_{n-1},x_n$ such that $x_0=x$, $x_n=y$ and $d(x_i,x_{i-1})\leq\min(R_{x_i}^n+\delta_n,R_{x_{i-1}}^n+\delta_n)$, for all $i=1,\ldots,n$.
Example: connected graph are $0$-connected spaces. If you like, I am just studying deformations of graphs.
Question: Suppose that $(X,d_n)$ is a sequence of $\delta_n$-connected spaces which converges in Gromov-Hausdorff sense to some $(X,d)$. Is it true that $\delta_n\rightarrow\delta$ and $(X,d)$ is $\delta$-connected?
Thank you in advance,
Valerio
