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Given a metric space $(\mathbb{A},d)$ 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.

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As it stands the question is incomprehensible. Maybe you should consult someone who can help you in formulating your question properly. –  Christian Blatter Aug 29 '12 at 15:24
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