An example shows that there are well-behaved infinte signaling games with no sequential equilibria. We explore the relationship between equilibrium outcomes of the infinite game and those of approximating games. Consider a sequence of signaling games approaching a limit game. A "(sub)sequence of equilibrium outcomes of the approximating games will converge to a limit distribution. That limit distribution will be an equilibrium outcome of the limit game if it can be realized by strategies of the limit game. As a result of this general convergence result, we prove the existance of sequential and weak-best-response equilibria for strongly monotonic games. In a companion article we explore the role of cheap talk in solving the non-existence problem.