Given a metric space $(\mathbb{A},d)$ in $\mathbb{R^n}$ with a metric $d$ being the Euclidean metric:

If $\lim_{t \rightarrow \infty}||A_{t+1}-A_t||\rightarrow 0$ is a convergent sequence where $A$ is a matrix with the rows being the points in the metric space and there exists a function $ f(.) $ acting on the rows of A that converges to the euclidean metric over the sequence as $\lim_{t \rightarrow \infty}||f_{i,j}(A_{t+1})-f_{i,j}(A_t)||\rightarrow d(A_{i.},A_{j.})$ where $i,j$ denote the rows of $A$ :-

am looking for some reference to study/characterize/understand this phenomenon of convergence of functions to a metric over a convergent sequence in a metric space