 A comment about the need for and usefulness of such convergence theorems is in order, now that we’ve seen one. Given some specific pair of scientific hypotheses \(h_i\) and \(h_j\) one may directly compute the likelihood, given \((h_{i}\cdot b\cdot c^{n})\), that a proposed sequence of experiments or observations \(c^n\) will result in one of the sequences of outcomes that would yield low likelihood ratios. So, given a specific pair of hypotheses and a proposed sequence of experiments, we don’t need a general Convergence Theorem to tell us the likelihood of obtaining refuting evidence. The specific hypotheses \(h_i\) and \(h_j\) tell us this themselves. They tell us the likelihood of obtaining each specific outcome stream, including those that either refute the competitor or produce a very small likelihood ratio for it. Furthermore, after we’ve actually performed an experiment and recorded its outcome, all that matters is the actual ratio of likelihoods for that outcome. Convergence theorems become moot.