The martingale convergence theorem implies that in certain situations a Bayesian agent is sure to converge to the truth in the limit. Gordon Belot has argued that this constitutes a liability for Bayesian epistemology since it ignores the many ways in which one might fail to identify truth in the limit. In this talk I will study convergence to the truth within a nonstandard probability framework that allows fine-grained distinctions between infinite hypotheses. Within the nonstandard framework, convergence to the truth is expected only for hypotheses that can be finitely approximated. Importantly, this leads not to a revision but a refinement of the standard martingale convergence theorem.