While it is often desired to compute the persistence diagram of a filtration of a topological space precisely, this is routinely not possible for many reasons. First, it might be impossible to encode the exact filtration into a computer. Second, even if this step is accomplished, it might be computationally infeasible to compute the associated persistence diagram (e.g. nerve lemma and Ae*ech complex). As a result, it is commonplace to substitute an approximation (e.g. Vietoris-Rips complex) and use its persistence diagram instead. But what, exactly, have we lost? Most computational topologists are aware of the bottleneck distance between two persistence diagrams, and the literature up until this point typically stops here in terms of error bounds of approximations (or perhaps goes one step further to log-bottleneck). In this talk, we propose a more rigorous framework for analyzing the approximations of persistence diagrams.