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We extend Kreps and Wilson's 1982 definition of sequential equilibrium to multi-stage games with infinite sets of signals and actions. We define “broad sequential epsilon-equilibria” by properties of “sequential epsilon-rationality” and “broad consistency.” Given beliefs, a player's strategy is sequentially epsilon-rational if, at every date t, at every possible signal outside a uniformly unlikely set, the player cannot expect to gain more than epsilon by any feasible deviation. Beliefs are broadly consistent if, with arbitrarily small perturbations of the strategies, all deviations from Bayes' rule become arbitrarily small on each open observable event. Broad sequential epsilon-equilibria are shown to exist for a large class of regular projective games. Examples illustrate properties of this solution and the difficulties of alternative approaches to the problem of extending sequential equilibrium to infinite games.